{"v":1,"topic":"M8","subject":"math","title":"Differentiation","testLen":20,"subs":[{"id":"d1","name":"Basic rules and first principles"},{"id":"d2","name":"Product and quotient rules"},{"id":"d3","name":"Chain rule"},{"id":"d4","name":"Trigonometric and inverse trig derivatives"},{"id":"d5","name":"Exponential and logarithmic derivatives"},{"id":"d6","name":"Implicit and parametric differentiation"},{"id":"d7","name":"Higher derivatives"},{"id":"d8","name":"Tangents, normals and rates"},{"id":"d9","name":"Maxima, minima and inflection"}],"learn":{"formulas":["$\\dfrac{d}{dx}x^n=nx^{n-1}$","$(uv)^{\\prime}=u^{\\prime}v+uv^{\\prime}$, $\\left(\\dfrac{u}{v}\\right)^{\\prime}=\\dfrac{u^{\\prime}v-uv^{\\prime}}{v^2}$","$\\dfrac{d}{dx}f(g(x))=f^{\\prime}(g(x))\\,g^{\\prime}(x)$","$(\\sin x)^{\\prime}=\\cos x$, $(\\cos x)^{\\prime}=-\\sin x$, $(\\tan x)^{\\prime}=\\sec^2x$, $(\\cot x)^{\\prime}=-\\csc^2x$","$(\\sec x)^{\\prime}=\\sec x\\tan x$, $(\\csc x)^{\\prime}=-\\csc x\\cot x$","$(\\sin^{-1}x)^{\\prime}=\\dfrac{1}{\\sqrt{1-x^2}}$, $(\\cos^{-1}x)^{\\prime}=-\\dfrac{1}{\\sqrt{1-x^2}}$, $(\\tan^{-1}x)^{\\prime}=\\dfrac{1}{1+x^2}$","$(e^x)^{\\prime}=e^x$, $(a^x)^{\\prime}=a^x\\ln a$, $(\\ln x)^{\\prime}=\\dfrac1x$, $(\\log_a x)^{\\prime}=\\dfrac{1}{x\\ln a}$","Parametric: $\\dfrac{dy}{dx}=\\dfrac{dy/dt}{dx/dt}$; second derivative: $\\dfrac{d}{dt}\\!\\left(\\dfrac{dy}{dx}\\right)\\div\\dfrac{dx}{dt}$","Max/min: $f^{\\prime}(c)=0$; $f^{\\prime\\prime}(c)<0$ max, $f^{\\prime\\prime}(c)>0$ min. Inflection: $f^{\\prime\\prime}$ changes sign."],"examples":[{"q":"Differentiate $y=\\sin^3(2x)$.","s":"$y^{\\prime}=3\\sin^2(2x)\\cdot\\cos(2x)\\cdot2=6\\sin^2 2x\\cos 2x$. Three layers, three factors."},{"q":"Find $\\dfrac{dy}{dx}$ if $x^3+y^3=6xy$.","s":"$3x^2+3y^2y^{\\prime}=6y+6xy^{\\prime}\\Rightarrow y^{\\prime}=\\dfrac{2y-x^2}{y^2-2x}$."},{"q":"Find the local extrema of $f(x)=2x^3-3x^2-12x$.","s":"$f^{\\prime}=6(x-2)(x+1)$. $f^{\\prime\\prime}(-1)=-18<0$: max $f(-1)=7$. $f^{\\prime\\prime}(2)=18>0$: min $f(2)=-20$."}],"traps":["Forgetting the inner derivative in the chain rule.","Treating $a^x$ or $e^{kx}$ like a power $x^n$.","Giving the location of a maximum when the question asks for the value, or the reverse.","Parametric second derivative: forgetting to divide by $dx/dt$ again.","A sign slip when differentiating $\\cos x$, $\\cot x$ or $\\cos^{-1}x$."]},"items":[{"id":"M8-0001","sub":"d1","tier":1,"skill":"recall","t":45,"fam":"M8-F0001","pool":"ladder","q":"$\\dfrac{d}{dx}\\left(x^5-3x^2+7\\right)=$","o":["$x^4-6x$","$5x^4-6x+7$","$5x^4-6x$","$5x^4-3x$"],"a":2,"ex":"Power rule term by term; the constant 7 differentiates to 0."},{"id":"M8-0002","sub":"d1","tier":1,"skill":"apply","t":45,"fam":"M8-F0002","pool":"ladder","q":"If $f(x)=\\sqrt{x}$, then $f^{\\prime}(4)=$","o":["$\\tfrac14$","$\\tfrac12$","$\\tfrac18$","$2$"],"a":0,"ex":"$f^{\\prime}(x)=\\dfrac{1}{2\\sqrt{x}}$, so $f^{\\prime}(4)=\\dfrac{1}{2\\cdot 2}=\\tfrac14$."},{"id":"M8-0003","sub":"d1","tier":1,"skill":"recall","t":45,"fam":"M8-F0003","pool":"ladder","q":"$\\dfrac{d}{dx}\\left(\\dfrac{1}{x^3}\\right)=$","o":["$-\\dfrac{1}{3x^2}$","$-\\dfrac{3}{x^4}$","$-\\dfrac{3}{x^2}$","$\\dfrac{3}{x^4}$"],"a":1,"ex":"Write as $x^{-3}$; the derivative is $-3x^{-4}$."},{"id":"M8-0004","sub":"d2","tier":1,"skill":"apply","t":45,"fam":"M8-F0004","pool":"ladder","q":"$\\dfrac{d}{dx}\\left(x^2\\sin x\\right)=$","o":["$2x\\sin x-x^2\\cos x$","$2x\\cos x$","$2x\\sin x+x^2\\cos x$","$x^2\\cos x$"],"a":2,"ex":"Product rule: $(x^2)^{\\prime}\\sin x+x^2(\\sin x)^{\\prime}$."},{"id":"M8-0005","sub":"d2","tier":1,"skill":"apply","t":45,"fam":"M8-F0005","pool":"ladder","q":"$\\dfrac{d}{dx}\\left(\\dfrac{x}{x+1}\\right)=$","o":["$\\dfrac{2x+1}{(x+1)^2}$","$-\\dfrac{1}{(x+1)^2}$","$\\dfrac{1}{(x+1)^2}$","$\\dfrac{1}{x+1}$"],"a":2,"ex":"Quotient rule: $\\dfrac{(x+1)\\cdot1-x\\cdot1}{(x+1)^2}=\\dfrac{1}{(x+1)^2}$."},{"id":"M8-0006","sub":"d3","tier":1,"skill":"apply","t":45,"fam":"M8-F0006","pool":"ladder","q":"$\\dfrac{d}{dx}(3x+1)^4=$","o":["$12(3x+1)^4$","$12(3x+1)^3$","$3(3x+1)^3$","$4(3x+1)^3$"],"a":1,"ex":"Chain rule: $4(3x+1)^3\\times 3$. Forgetting the inner derivative gives $4(3x+1)^3$."},{"id":"M8-0007","sub":"d3","tier":1,"skill":"recall","t":45,"fam":"M8-F0007","pool":"ladder","q":"$\\dfrac{d}{dx}\\sin 5x=$","o":["$-5\\cos 5x$","$5\\cos 5x$","$\\tfrac15\\cos 5x$","$\\cos 5x$"],"a":1,"ex":"Chain rule multiplies by the inner derivative 5."},{"id":"M8-0008","sub":"d4","tier":1,"skill":"recall","t":45,"fam":"M8-F0008","pool":"ladder","q":"$\\dfrac{d}{dx}\\tan x=$","o":["$-\\sec^2 x$","$\\sec^2 x$","$\\sec x\\tan x$","$-\\csc^2 x$"],"a":1,"ex":"Standard result; $-\\csc^2 x$ is the derivative of $\\cot x$."},{"id":"M8-0009","sub":"d4","tier":1,"skill":"recall","t":45,"fam":"M8-F0009","pool":"ladder","q":"$\\dfrac{d}{dx}\\sin^{-1}x=$","o":["$-\\dfrac{1}{\\sqrt{1-x^2}}$","$\\dfrac{1}{1+x^2}$","$\\dfrac{1}{\\sqrt{1-x^2}}$","$\\dfrac{1}{x\\sqrt{x^2-1}}$"],"a":2,"ex":"Standard result; the negative version belongs to $\\cos^{-1}x$."},{"id":"M8-0010","sub":"d5","tier":1,"skill":"recall","t":45,"fam":"M8-F0010","pool":"ladder","q":"$\\dfrac{d}{dx}e^{2x}=$","o":["$2e^{2x}$","$e^{2x}$","$2x\\,e^{2x-1}$","$\\tfrac12 e^{2x}$"],"a":0,"ex":"Chain rule on $e^{u}$ with $u=2x$. Treating $e^{2x}$ like a power gives the wrong $2x\\,e^{2x-1}$."},{"id":"M8-0011","sub":"d5","tier":1,"skill":"apply","t":45,"fam":"M8-F0011","pool":"ladder","q":"$\\dfrac{d}{dx}\\ln(3x)=$","o":["$\\dfrac{\\ln 3}{x}$","$\\dfrac{1}{3x}$","$\\dfrac{3}{x}$","$\\dfrac{1}{x}$"],"a":3,"ex":"$\\dfrac{1}{3x}\\times 3=\\dfrac1x$. Equivalently $\\ln 3x=\\ln3+\\ln x$."},{"id":"M8-0012","sub":"d6","tier":1,"skill":"apply","t":45,"fam":"M8-F0012","pool":"ladder","q":"If $x^2+y^2=25$, then $\\dfrac{dy}{dx}=$","o":["$\\dfrac{x}{y}$","$-2x$","$-\\dfrac{x}{y}$","$-\\dfrac{y}{x}$"],"a":2,"ex":"Differentiate implicitly: $2x+2y\\,y^{\\prime}=0$."},{"id":"M8-0013","sub":"d7","tier":1,"skill":"apply","t":45,"fam":"M8-F0013","pool":"ladder","q":"If $y=x^4$, then $\\dfrac{d^2y}{dx^2}=$","o":["$12x$","$4x^3$","$24x$","$12x^2$"],"a":3,"ex":"$y^{\\prime}=4x^3$, $y^{\\prime\\prime}=12x^2$."},{"id":"M8-0014","sub":"d8","tier":1,"skill":"apply","t":45,"fam":"M8-F0014","pool":"ladder","q":"The slope of the tangent to $y=x^2+3x$ at $x=1$ is","o":["$5$","$4$","$2$","$3$"],"a":0,"ex":"$y^{\\prime}=2x+3=5$ at $x=1$. The value 4 is $y(1)$, not the slope."},{"id":"M8-0015","sub":"d9","tier":1,"skill":"apply","t":45,"fam":"M8-F0015","pool":"ladder","q":"$f(x)=x^2-4x+1$ has its minimum at $x=$","o":["$4$","$2$","$-3$","$-2$"],"a":1,"ex":"$f^{\\prime}(x)=2x-4=0$ gives $x=2$; $-3$ is the minimum value, not its location."},{"id":"M8-0016","sub":"d1","tier":1,"skill":"apply","t":45,"fam":"M8-F0016","pool":"ladder","q":"If $f(x)=2x^3$, then $f^{\\prime}(1)=$","o":["$12$","$6$","$2$","$3$"],"a":1,"ex":"$f^{\\prime}(x)=6x^2$, so $f^{\\prime}(1)=6$; 12 is $f^{\\prime\\prime}(1)$."},{"id":"M8-0017","sub":"d4","tier":1,"skill":"apply","t":45,"fam":"M8-F0017","pool":"ladder","q":"$\\left.\\dfrac{d}{dx}\\cos x\\right|_{x=\\pi/2}=$","o":["$1$","$\\dfrac{\\pi}{2}$","$0$","$-1$"],"a":3,"ex":"$\\dfrac{d}{dx}\\cos x=-\\sin x$, and $-\\sin\\tfrac{\\pi}{2}=-1$."},{"id":"M8-0018","sub":"d5","tier":1,"skill":"recall","t":45,"fam":"M8-F0018","pool":"ladder","q":"For $a>0$, $\\dfrac{d}{dx}a^x=$","o":["$x\\,a^{x-1}$","$a^x$","$\\dfrac{a^x}{\\ln a}$","$a^x\\ln a$"],"a":3,"ex":"$a^x=e^{x\\ln a}$, so the derivative is $a^x\\ln a$. $x\\,a^{x-1}$ wrongly treats it as a power."},{"id":"M8-0019","sub":"d3","tier":1,"skill":"apply","t":45,"fam":"M8-F0019","pool":"ladder","q":"$\\dfrac{d}{dx}\\sqrt{x^2+1}=$","o":["$\\dfrac{1}{2\\sqrt{x^2+1}}$","$x\\sqrt{x^2+1}$","$\\dfrac{2x}{\\sqrt{x^2+1}}$","$\\dfrac{x}{\\sqrt{x^2+1}}$"],"a":3,"ex":"$\\dfrac{1}{2\\sqrt{x^2+1}}\\times 2x$."},{"id":"M8-0020","sub":"d7","tier":1,"skill":"apply","t":45,"fam":"M8-F0020","pool":"ladder","q":"If $y=\\sin x$, then $y^{\\prime\\prime}+y=$","o":["$2\\cos x$","$2\\sin x$","$0$","$-2\\sin x$"],"a":2,"ex":"$y^{\\prime\\prime}=-\\sin x$, so $y^{\\prime\\prime}+y=0$."},{"id":"M8-0021","sub":"d2","tier":2,"skill":"multi","t":60,"fam":"M8-F0021","pool":"ladder","q":"$\\dfrac{d}{dx}\\left(x\\,e^{-x}\\right)=$","o":["$e^{-x}(1-x)$","$e^{-x}(x-1)$","$e^{-x}(1+x)$","$-x\\,e^{-x}$"],"a":0,"ex":"Product rule: $e^{-x}+x(-e^{-x})$. Missing the minus from the chain rule gives $e^{-x}(1+x)$."},{"id":"M8-0022","sub":"d4","tier":2,"skill":"multi","t":60,"fam":"M8-F0022","pool":"ladder","q":"$\\dfrac{d}{dx}\\tan^{-1}(2x)=$","o":["$\\dfrac{2}{1+4x^2}$","$\\dfrac{2}{1+2x^2}$","$\\dfrac{1}{1+4x^2}$","$\\dfrac{1}{1+2x^2}$"],"a":0,"ex":"$\\dfrac{1}{1+(2x)^2}\\times 2$. Two errors to avoid: squaring only $x$, and dropping the factor 2."},{"id":"M8-0023","sub":"d6","tier":2,"skill":"multi","t":60,"fam":"M8-F0023","pool":"ladder","q":"For the curve $xy=4$, $\\dfrac{dy}{dx}$ at $(2,2)$ is","o":["$1$","$0$","$-2$","$-1$"],"a":3,"ex":"$y+xy^{\\prime}=0\\Rightarrow y^{\\prime}=-\\dfrac{y}{x}=-1$."},{"id":"M8-0024","sub":"d6","tier":2,"skill":"multi","t":60,"fam":"M8-F0024","pool":"ladder","q":"If $x=t^2,\\ y=t^3$, then $\\dfrac{dy}{dx}=$","o":["$\\dfrac{3t^2}{2}$","$\\dfrac{2}{3t}$","$3t^2$","$\\dfrac{3t}{2}$"],"a":3,"ex":"$\\dfrac{dy/dt}{dx/dt}=\\dfrac{3t^2}{2t}=\\dfrac{3t}{2}$. Inverting the ratio gives $\\dfrac{2}{3t}$."},{"id":"M8-0025","sub":"d5","tier":2,"skill":"multi","t":60,"fam":"M8-F0025","pool":"ladder","q":"$\\dfrac{d}{dx}\\ln(\\sin x)=$","o":["$\\cos x\\,\\ln(\\sin x)$","$\\cot x$","$\\tan x$","$\\dfrac{1}{\\sin x}$"],"a":1,"ex":"$\\dfrac{1}{\\sin x}\\cdot\\cos x=\\cot x$."},{"id":"M8-0026","sub":"d5","tier":2,"skill":"multi","t":60,"fam":"M8-F0026","pool":"ladder","q":"For $x>0$, $\\dfrac{d}{dx}x^x=$","o":["$x^x\\ln x$","$x\\cdot x^{x-1}$","$x^x+\\ln x$","$x^x(1+\\ln x)$"],"a":3,"ex":"Take logs: $\\ln y=x\\ln x\\Rightarrow \\dfrac{y^{\\prime}}{y}=\\ln x+1$. Neither the power rule nor the exponential rule alone applies."},{"id":"M8-0027","sub":"d8","tier":2,"skill":"multi","t":60,"fam":"M8-F0027","pool":"ladder","q":"The equation of the normal to $y=x^2$ at $(1,1)$ is","o":["$2x-y=1$","$x+2y=3$","$2x+y=3$","$x-2y=-1$"],"a":1,"ex":"Tangent slope is 2, so the normal slope is $-\\tfrac12$: $y-1=-\\tfrac12(x-1)$. $2x-y=1$ is the tangent."},{"id":"M8-0028","sub":"d9","tier":2,"skill":"multi","t":60,"fam":"M8-F0028","pool":"ladder","q":"The maximum value of $f(x)=x(6-x)$ is","o":["$6$","$36$","$9$","$3$"],"a":2,"ex":"$f^{\\prime}=6-2x=0$ at $x=3$, so $f(3)=9$. 3 is where the maximum occurs."},{"id":"M8-0029","sub":"d7","tier":2,"skill":"multi","t":60,"fam":"M8-F0029","pool":"ladder","q":"If $y=e^{kx}$ satisfies $y^{\\prime\\prime}-3y^{\\prime}+2y=0$, then $k=$","o":["$2$ only","$-1$ or $-2$","$1$ or $-2$","$1$ or $2$"],"a":3,"ex":"Substituting gives $k^2-3k+2=0$, so $k=1$ or $2$."},{"id":"M8-0030","sub":"d1","tier":2,"skill":"apply","t":60,"fam":"M8-F0030","pool":"ladder","q":"$\\displaystyle\\lim_{h\\to0}\\frac{(2+h)^3-8}{h}=$","o":["$12$","$0$","$8$","$6$"],"a":0,"ex":"This is $f^{\\prime}(2)$ for $f(x)=x^3$: $3(2)^2=12$."},{"id":"M8-0031","sub":"d3","tier":2,"skill":"multi","t":60,"fam":"M8-F0031","pool":"ladder","q":"$\\dfrac{d}{dx}\\sin^2(3x)=$","o":["$6\\sin 6x$","$3\\sin 6x$","$\\sin 6x$","$6\\sin 3x$"],"a":1,"ex":"$2\\sin3x\\cdot\\cos3x\\cdot3=3\\sin6x$. Dropping the inner factor 3 gives $\\sin 6x$."},{"id":"M8-0032","sub":"d8","tier":2,"skill":"multi","t":60,"fam":"M8-F0032","pool":"ladder","q":"$f(x)=x^3-3x$ is increasing for","o":["$-1<x<1$","all real $x$","$x>0$ only","$x<-1$ or $x>1$"],"a":3,"ex":"$f^{\\prime}(x)=3(x^2-1)>0$ when $|x|>1$."},{"id":"M8-0033","sub":"d4","tier":2,"skill":"multi","t":60,"fam":"M8-F0033","pool":"ladder","q":"$\\dfrac{d}{dx}\\left(\\sin^{-1}x+\\cos^{-1}x\\right)=$","o":["$-\\dfrac{2}{\\sqrt{1-x^2}}$","$\\dfrac{2}{\\sqrt{1-x^2}}$","$\\dfrac{\\pi}{2}$","$0$"],"a":3,"ex":"The sum is the constant $\\tfrac{\\pi}{2}$, so its derivative is 0. $\\tfrac{\\pi}{2}$ is the value, not the derivative."},{"id":"M8-0034","sub":"d2","tier":2,"skill":"multi","t":60,"fam":"M8-F0034","pool":"ladder","q":"$\\dfrac{d}{dx}\\left(\\dfrac{1-x}{1+x}\\right)=$","o":["$-\\dfrac{1}{(1+x)^2}$","$-\\dfrac{2x}{(1+x)^2}$","$\\dfrac{2}{(1+x)^2}$","$-\\dfrac{2}{(1+x)^2}$"],"a":3,"ex":"$\\dfrac{(1+x)(-1)-(1-x)(1)}{(1+x)^2}=\\dfrac{-2}{(1+x)^2}$."},{"id":"M8-0035","sub":"d9","tier":2,"skill":"multi","t":60,"fam":"M8-F0035","pool":"ladder","q":"Two positive numbers have sum 20. Their maximum possible product is","o":["$200$","$400$","$96$","$100$"],"a":3,"ex":"$P=x(20-x)$, $P^{\\prime}=0$ at $x=10$, so $P=100$."},{"id":"M8-0036","sub":"d6","tier":2,"skill":"multi","t":60,"fam":"M8-F0036","pool":"ladder","q":"For $x^2+xy+y^2=7$, $\\dfrac{dy}{dx}$ at $(1,2)$ is","o":["$-\\dfrac45$","$\\dfrac45$","$-\\dfrac54$","$-1$"],"a":0,"ex":"$2x+y+xy^{\\prime}+2yy^{\\prime}=0\\Rightarrow y^{\\prime}=-\\dfrac{2x+y}{x+2y}=-\\dfrac45$."},{"id":"M8-0037","sub":"d5","tier":2,"skill":"apply","t":60,"fam":"M8-F0037","pool":"ladder","q":"$\\dfrac{d}{dx}\\log_{10}x=$","o":["$\\dfrac{1}{x}$","$\\dfrac{1}{x\\ln 10}$","$\\dfrac{\\ln 10}{x}$","$\\dfrac{10}{x}$"],"a":1,"ex":"$\\log_{10}x=\\dfrac{\\ln x}{\\ln 10}$."},{"id":"M8-0038","sub":"d8","tier":2,"skill":"apply","t":60,"fam":"M8-F0038","pool":"ladder","q":"The rate of change of the area of a circle with respect to its radius, when $r=3$, is","o":["$3\\pi$","$12\\pi$","$6\\pi$","$9\\pi$"],"a":2,"ex":"$\\dfrac{dA}{dr}=2\\pi r=6\\pi$. $9\\pi$ is the area itself."},{"id":"M8-0039","sub":"d7","tier":2,"skill":"multi","t":60,"fam":"M8-F0039","pool":"ladder","q":"If $y=\\ln x$, then $\\dfrac{d^3y}{dx^3}=$","o":["$\\dfrac{2}{x^3}$","$\\dfrac{6}{x^4}$","$-\\dfrac{1}{x^2}$","$-\\dfrac{2}{x^3}$"],"a":0,"ex":"$\\tfrac1x\\to-\\tfrac{1}{x^2}\\to\\tfrac{2}{x^3}$. Watch the sign flip at each step."},{"id":"M8-0040","sub":"d3","tier":2,"skill":"apply","t":60,"fam":"M8-F0040","pool":"ladder","q":"$\\dfrac{d}{dx}e^{\\sin x}=$","o":["$e^{\\cos x}$","$\\sin x\\,e^{\\sin x-1}$","$e^{\\sin x}$","$\\cos x\\,e^{\\sin x}$"],"a":3,"ex":"Chain rule: $e^{\\sin x}\\cdot\\cos x$."},{"id":"M8-0041","sub":"d9","tier":3,"skill":"multi","t":75,"fam":"M8-F0041","pool":"ladder","q":"The local maximum value of $f(x)=x^3-6x^2+9x+1$ is","o":["$1$","$3$","$5$","$9$"],"a":2,"ex":"$f^{\\prime}=3(x-1)(x-3)$; $f^{\\prime\\prime}(1)<0$, so the local max is $f(1)=5$. $f(3)=1$ is the local min."},{"id":"M8-0042","sub":"d7","tier":3,"skill":"multi","t":75,"fam":"M8-F0042","pool":"ladder","q":"If $x^2+y^2=1$, then $\\dfrac{d^2y}{dx^2}=$","o":["$-\\dfrac{x}{y^2}$","$\\dfrac{1}{y^3}$","$-\\dfrac{1}{y^3}$","$-\\dfrac{1}{y^2}$"],"a":2,"ex":"$y^{\\prime}=-\\tfrac{x}{y}$, so $y^{\\prime\\prime}=-\\dfrac{y-xy^{\\prime}}{y^2}=-\\dfrac{y^2+x^2}{y^3}=-\\dfrac{1}{y^3}$ using the curve."},{"id":"M8-0043","sub":"d5","tier":3,"skill":"multi","t":75,"fam":"M8-F0043","pool":"ladder","q":"For $x>0$, $\\dfrac{d}{dx}\\,x^{\\ln x}=$","o":["$2\\ln x\\cdot x^{\\ln x}$","$\\ln x\\cdot x^{\\ln x-1}$","$2x^{\\ln x-1}\\ln x$","$\\dfrac{x^{\\ln x}}{x}$"],"a":2,"ex":"$\\ln y=(\\ln x)^2\\Rightarrow\\dfrac{y^{\\prime}}{y}=\\dfrac{2\\ln x}{x}$."},{"id":"M8-0044","sub":"d1","tier":3,"skill":"trap","t":75,"fam":"M8-F0044","pool":"ladder","q":"At $x=0$, the function $f(x)=|x|$ is","o":["continuous but not differentiable","differentiable but not continuous","neither continuous nor differentiable","both continuous and differentiable"],"a":0,"ex":"The graph has no break, but the left derivative is $-1$ and the right derivative is $+1$."},{"id":"M8-0045","sub":"d8","tier":3,"skill":"multi","t":75,"fam":"M8-F0045","pool":"ladder","q":"Tangents to $y=x^3$ that are parallel to $y=12x+1$ touch the curve at","o":["$x=2$ and $x=-2$","$x=2$ only","$x=\\pm\\sqrt{12}$","$x=4$"],"a":0,"ex":"$3x^2=12\\Rightarrow x=\\pm2$. Two points, so \"$x=2$ only\" misses one."},{"id":"M8-0046","sub":"d4","tier":3,"skill":"multi","t":75,"fam":"M8-F0046","pool":"ladder","q":"For $|x|<1$, $\\dfrac{d}{dx}\\tan^{-1}\\!\\left(\\dfrac{2x}{1-x^2}\\right)=$","o":["$\\dfrac{2}{1+x^2}$","$\\dfrac{2x}{1+x^2}$","$-\\dfrac{2}{1+x^2}$","$\\dfrac{1}{1+x^2}$"],"a":0,"ex":"Put $x=\\tan\\theta$: the expression is $\\tan^{-1}(\\tan2\\theta)=2\\tan^{-1}x$, whose derivative is $\\dfrac{2}{1+x^2}$."},{"id":"M8-0047","sub":"d9","tier":3,"skill":"multi","t":75,"fam":"M8-F0047","pool":"ladder","q":"Equal squares of side $x$ cm are cut from the corners of a $12\\times12$ cm sheet to make an open box. The volume is greatest when $x=$","o":["$2$","$4$","$6$","$3$"],"a":0,"ex":"$V=x(12-2x)^2$, $V^{\\prime}=(12-2x)(12-6x)=0$. $x=6$ gives zero volume, so $x=2$."},{"id":"M8-0048","sub":"d2","tier":3,"skill":"multi","t":75,"fam":"M8-F0048","pool":"ladder","q":"$\\dfrac{d}{dx}\\left[e^x(\\sin x+\\cos x)\\right]=$","o":["$2e^x\\sin x$","$e^x\\cos x$","$2e^x\\cos x$","$2e^x(\\sin x+\\cos x)$"],"a":2,"ex":"$e^x(\\sin x+\\cos x)+e^x(\\cos x-\\sin x)=2e^x\\cos x$."},{"id":"M8-0049","sub":"d6","tier":3,"skill":"multi","t":75,"fam":"M8-F0049","pool":"ladder","q":"If $x=a\\cos t,\\ y=a\\sin t$, then $\\dfrac{d^2y}{dx^2}=$","o":["$-\\cot t$","$\\dfrac{1}{a\\sin^3 t}$","$-\\csc^2 t$","$-\\dfrac{1}{a\\sin^3 t}$"],"a":3,"ex":"$\\dfrac{dy}{dx}=-\\cot t$. Then divide $\\dfrac{d}{dt}(-\\cot t)=\\csc^2t$ by $\\dfrac{dx}{dt}=-a\\sin t$. Stopping at $\\csc^2 t$ is the classic slip."},{"id":"M8-0050","sub":"d8","tier":3,"skill":"multi","t":75,"fam":"M8-F0050","pool":"ladder","q":"$f(x)=x^2e^{-x}$ is decreasing for","o":["$x<0$ or $x>2$","$0<x<2$","$x<0$ only","$x>2$ only"],"a":0,"ex":"$f^{\\prime}(x)=x(2-x)e^{-x}$, which is negative outside $[0,2]$."},{"id":"M8-0051","sub":"d7","tier":3,"skill":"apply","t":75,"fam":"M8-F0051","pool":"ladder","q":"If $y=\\sin 2x$, then $\\dfrac{d^4y}{dx^4}=$","o":["$16\\cos 2x$","$16\\sin 2x$","$8\\cos 2x$","$-16\\sin 2x$"],"a":1,"ex":"Each derivative multiplies by 2 and cycles $\\sin\\to\\cos\\to-\\sin\\to-\\cos\\to\\sin$."},{"id":"M8-0052","sub":"d3","tier":3,"skill":"multi","t":75,"fam":"M8-F0052","pool":"ladder","q":"$\\dfrac{d}{dx}\\ln\\!\\left(x+\\sqrt{x^2+1}\\right)=$","o":["$\\dfrac{1}{\\sqrt{x^2+1}}$","$\\dfrac{1}{x+\\sqrt{x^2+1}}$","$\\dfrac{x}{\\sqrt{x^2+1}}$","$1+\\dfrac{x}{\\sqrt{x^2+1}}$"],"a":0,"ex":"$\\dfrac{1+\\frac{x}{\\sqrt{x^2+1}}}{x+\\sqrt{x^2+1}}=\\dfrac{1}{\\sqrt{x^2+1}}$ after simplifying the numerator."},{"id":"M8-0053","sub":"d9","tier":3,"skill":"trap","t":75,"fam":"M8-F0053","pool":"ladder","q":"For $x>0$, the minimum value of $x+\\dfrac1x$ is","o":["$2$","it has no minimum","$1$","$0$"],"a":0,"ex":"$1-\\tfrac{1}{x^2}=0$ at $x=1$; the value there is 2. 1 is where it occurs."},{"id":"M8-0054","sub":"d1","tier":3,"skill":"trap","t":75,"fam":"M8-F0054","pool":"ladder","q":"If $f(x)=x|x|$, then $f^{\\prime}(0)=$","o":["$-1$","$0$","does not exist","$1$"],"a":1,"ex":"$\\dfrac{f(h)-f(0)}{h}=|h|\\to0$ from both sides. Unlike $|x|$, this function is differentiable at 0."},{"id":"M8-0055","sub":"d6","tier":3,"skill":"trap","t":75,"fam":"M8-F0055","pool":"ladder","q":"If $y=\\sqrt{x+\\sqrt{x+\\sqrt{x+\\cdots}}}$, then $\\dfrac{dy}{dx}=$","o":["$\\dfrac{1}{y-1}$","$\\dfrac{1}{2y-1}$","$2y-1$","$\\dfrac{1}{2y}$"],"a":1,"ex":"$y^2=x+y\\Rightarrow 2yy^{\\prime}=1+y^{\\prime}\\Rightarrow y^{\\prime}=\\dfrac{1}{2y-1}$."},{"id":"M8-0056","sub":"d8","tier":3,"skill":"apply","t":75,"fam":"M8-F0056","pool":"ladder","q":"Using differentials, $\\sqrt{25.1}\\approx$","o":["$5.02$","$5.01$","$5.001$","$5.1$"],"a":1,"ex":"$\\Delta y\\approx\\dfrac{1}{2\\sqrt{25}}(0.1)=0.01$."},{"id":"M8-0057","sub":"d5","tier":3,"skill":"multi","t":75,"fam":"M8-F0057","pool":"ladder","q":"For $|x|<1$, $\\dfrac{d}{dx}\\ln\\!\\left(\\dfrac{1+x}{1-x}\\right)=$","o":["$\\dfrac{2x}{1-x^2}$","$\\dfrac{2}{1-x^2}$","$\\dfrac{1}{1-x^2}$","$-\\dfrac{2}{1-x^2}$"],"a":1,"ex":"Split: $\\ln(1+x)-\\ln(1-x)$ gives $\\dfrac{1}{1+x}+\\dfrac{1}{1-x}$."},{"id":"M8-0058","sub":"d9","tier":3,"skill":"apply","t":75,"fam":"M8-F0058","pool":"ladder","q":"The point of inflection of $y=x^3-3x^2+4$ is at $x=$","o":["$0$","$2$","$1$","$3$"],"a":2,"ex":"$y^{\\prime\\prime}=6x-6=0$ at $x=1$, and it changes sign there. 0 and 2 are the turning points."},{"id":"M8-0059","sub":"d2","tier":3,"skill":"multi","t":75,"fam":"M8-F0059","pool":"ladder","q":"$\\dfrac{d}{dx}\\left(x\\sin^{-1}x\\right)=$","o":["$\\dfrac{x}{\\sqrt{1-x^2}}$","$\\sin^{-1}x-\\dfrac{x}{\\sqrt{1-x^2}}$","$\\sin^{-1}x+\\dfrac{x}{\\sqrt{1-x^2}}$","$\\dfrac{1}{\\sqrt{1-x^2}}$"],"a":2,"ex":"Product rule with $\\dfrac{d}{dx}\\sin^{-1}x=\\dfrac{1}{\\sqrt{1-x^2}}$."},{"id":"M8-0060","sub":"d7","tier":3,"skill":"multi","t":75,"fam":"M8-F0060","pool":"ladder","q":"If $y=x^2e^x$, then $y^{\\prime\\prime}(0)=$","o":["$0$","$1$","$2$","$4$"],"a":2,"ex":"$y^{\\prime\\prime}=e^x(x^2+4x+2)$, which is 2 at $x=0$."}]}