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AP Calculus AB : Derivatives

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Example Questions

Example Question #21 : Implicit Differentiation

Given that y=y(x) , find the derivative of the function

$y=\sin(xy)+y^3$

Possible Answers:

$y'=\frac{y\cos(xy)}{1-x\cos(xy)-3y^3}$

$y'=\frac{y\cos(xy)}{1-x\sin(xy)-3y^2}$

$y'=\frac{y\cos(xy)}{1-x\cos(xy)-3y^2}$

$y'=\frac{x\cos(xy)}{1-x\cos(xy)-3y^2}$

Correct answer:

$y'=\frac{y\cos(xy)}{1-x\cos(xy)-3y^2}$

Explanation:

To find the derivative with respect to y, we use implicit differentiation, which is an application of the chain rule.

$y=\sin(xy)+y^3$

$y'=\cos(xy)(y+xy')+3y^2y'$

$y'-xy'\cos(xy)-3y^2y'=y\cos(xy)$

$y'(1-x\cos(xy)-3y^2)=y\cos(xy)$

$y'=\frac{y\cos(xy)}{1-x\cos(xy)-3y^2}$

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Example Question #22 : Implicit Differentiation

Given that y=y(x) , find the derivative of the function

$y=\tan(y^2)+xy^5$

Possible Answers:

$y'=\frac{5y^4}{1-2y\sec^2(y^2)-5xy^4}$

$y'=\frac{y^5}{1-2y\sec^2(y^2)-5xy^4}$

$y'=\frac{y^5}{1+2y\sec^2(y^2)+5xy^4}$

$y'=\frac{y^5}{1-2y\tan^2(y^2)-5xy^4}$

Correct answer:

$y'=\frac{y^5}{1-2y\sec^2(y^2)-5xy^4}$

Explanation:

To find the derivative with respect to y, we use implicit differentiation, which is an application of the chain rule

$y=\tan(y^2)+xy^5$

$y'=\sec^2(y^2)(2yy')+y^5+5xy^4(y')$

$y'-2yy'\sec^2(y^2)-5xy^4y'=y^5$

$y'(1-2y\sec^2(y^2)-5xy^4)=y^5$

$y'=\frac{y^5}{1-2y\sec^2(y^2)-5xy^4}$

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Example Question #23 : Implicit Differentiation

Given that y=y(x) , find the derivative of the function

$y=2xy^3+x\cos(y)$

Possible Answers:

$y'=\frac{2y^3+\cos(y)}{1-x\sin(y)-6xy^2}$

$y'=\frac{2y^3+\sin(y)}{1+x\sin(y)-6xy^2}$

$y'=\frac{2y^3+\cos(y)}{1+x\sin(y)-6xy^2}$

$y'=\frac{2y^3+\cos(y)}{1+\sin(xy)-6xy^2}$

Correct answer:

$y'=\frac{2y^3+\cos(y)}{1+x\sin(y)-6xy^2}$

Explanation:

To find the derivative with respect to y, we use implicit differentiation, which is an application of the chain rule

$y=2xy^3+x\cos(y)$

$y'=2y^3+6xy^2y'+\cos(y)-x\sin(y)y'$

$y'+xy'\sin(y)-6xy^2y'=2y^3+\cos(y)$

$y'(1+x\sin(y)-6xy^2)=2y^3+\cos(y)$

$y'=\frac{2y^3+\cos(y)}{1+x\sin(y)-6xy^2}$

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Example Question #21 : Implicit Differentiation

Find $\frac{\mathrm{d} y}{\mathrm{d} x}$ :

$xy+\ln(y)=x$

Possible Answers:

$\frac{y-y^2}{xy+1}$

$\frac{y^2-y}{x}$

$\frac{y-y^2}{xy+y}$

$\frac{-y}{xy+1}$

Correct answer:

$\frac{y-y^2}{xy+1}$

Explanation:

To find $\frac{\mathrm{d} y}{\mathrm{d} x}$ we must use implicit differentiation, which is an application of the chain rule.
Taking $\frac{\mathrm{d} y}{\mathrm{d} x}$ of both sides of the equation, we get

$y+x\frac{\mathrm{d} y}{\mathrm{d} x}+\frac{1}{y}\frac{\mathrm{d} y}{\mathrm{d} x}=1$

The derivative was found using the following rules:

$\frac{\mathrm{d} }{\mathrm{d} x} f(g(x))=f'(g(x))\cdot g'(x)$ , $\frac{\mathrm{d} }{\mathrm{d} x} ax=a$ , $\frac{\mathrm{d} }{\mathrm{d} x} \ln(x)=\frac{1}{x}$

Note that for every derivative of a function with y, the additional term $\frac{\mathrm{d} y}{\mathrm{d} x}$ appears; this is because of the chain rule, where y=g(x) , so to speak, for the function it appears in.

Using algebra to solve for $\frac{\mathrm{d} y}{\mathrm{d} x}$ , we get

$\frac{\mathrm{d} y}{\mathrm{d} x}= \frac{1-y}{x+\frac{1}{y}}= \frac{y-y^2}{xy+1}$

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Example Question #27 : Implicit Differentiation

Find $\frac{\mathrm{d} y}{\mathrm{d} x}$ :

$3y+x\tan(y)=2$

Possible Answers:

$\frac{\tan(y)}{3+x\sec^2(y)}$

$\frac{-\tan(y)}{3-x\sec^2(y)}$

$\frac{-\tan(y)}{3+x\sec^2(y)}$

$\frac{-\tan(y)}{3+\sec^2(y)}$

Correct answer:

$\frac{-\tan(y)}{3+x\sec^2(y)}$

Explanation:

To find $\frac{\mathrm{d} y}{\mathrm{d} x}$ we must use implicit differentiation, which is an application of the chain rule.
Taking $\frac{\mathrm{d} }{\mathrm{d} x}$ of both sides of the equation, we get

$3 \frac{\mathrm{d} y}{\mathrm{d} x} +\tan(y)+x\sec^2(y)\frac{\mathrm{d} y}{\mathrm{d} x}=0$

The following derivative rules were used:

$\frac{\mathrm{d} }{\mathrm{d} x} f(g(x))=f'(g(x))\cdot g'(x)$ , $\frac{\mathrm{d} }{\mathrm{d} x} ax=a$ , $\frac{\mathrm{d} }{\mathrm{d} x} f(x)g(x)=f'(x)g(x)+f(x)g'(x)$ , $\frac{\mathrm{d} }{\mathrm{d} x} \tan(x)=\sec^2(x)$

Using algebra to solve for $\frac{\mathrm{d} y}{\mathrm{d} x}$ , we get

$\frac{\mathrm{d} y}{\mathrm{d} x}=\frac{-\tan(y)}{3+x\sec^2(y)}$

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Example Question #81 : Applications Of Derivatives

Find $\frac{\mathrm{d} y}{\mathrm{d} x}$ :

x^2+xy=4

Possible Answers:

$\frac{2x-y}{x}$

$\frac{-2x-y}{x}$

-2x-y

$\frac{1}{x}$

Correct answer:

$\frac{-2x-y}{x}$

Explanation:

$2x+y+x\frac{\mathrm{d} y}{\mathrm{d} x}=0$

using the following rules:

Using algebra to solve for $\frac{\mathrm{d} y}{\mathrm{d} x}$ , we get

$\frac{-2x-y}{x}$ .

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Example Question #21 : Implicit Differentiation

Use implicit differentiation to calculate the equation of the line tangent to the equation $2x^{2}+y^{2}=9$ at the point (2,1).

Possible Answers:

y=4x+9

y=-4x-9

y=-4x+9

y=4x-9

Correct answer:

y=-4x+9

Explanation:

Differentiate both sides of the equation: $\frac{\mathrm{d} }{\mathrm{d} x}(2x^{2}+y^{2}=9)$

Simplify: $4x+\frac{\mathrm{d} }{\mathrm{d} x}y^{2}=0$

Use implicit differentiation to differentiate the y term: $4x+2y\frac{dy}{dx}=0$

Subtract 4x from both sides of the equation: $2y\frac{dy}{dx}=-4x$

Divide both sides of the equation by 2y: $\frac{dy}{dx}=\frac{-4x}{2y}=\frac{-2x}{y}$

Plug in the appropriate values for x and y to find the slope of the tangent line: $m=\frac{-2(2)}{(1)}=-4$

Use slope-intercept form to solve for the equation of the tangent line: y=mx+b=y=-4x+b

Plug in the appropriate values of x and y into the equation, to find the equation of the tangent line: 1=(2)(-4)+b

Solve for b: 1=-8+b

b=9

Solution: y=-4x+9

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Example Question #21 : Implicit Differentiation

Find $\frac{\mathrm{d} z}{\mathrm{d} x}$ , where is a function of x.

Possible Answers:

$\frac{3-z}{x+2z}$

$\frac{3-2z}{x}$

$\frac{-z}{x+2z}$

Correct answer:

$\frac{3-z}{x+2z}$

Explanation:

To find $\frac{\mathrm{d} z}{\mathrm{d} x}$ we must use implicit differentiation, which is an application of the chain rule.
Taking $\frac{\mathrm{d} }{\mathrm{d} x}$ of both sides of the equation, we get

$z+x\frac{\mathrm{d} z}{\mathrm{d} x}+2z\frac{\mathrm{d} z}{\mathrm{d} x}=3$

and the derivatives were found using the following rules:

$\frac{\mathrm{d} }{\mathrm{d} x} f(g(x))=f'(g(x))\cdot g'(x)$ , $\frac{\mathrm{d} }{\mathrm{d} x} f(x)g(x)=f'(x)g(x)+f(x)g'(x)$ , $\frac{\mathrm{d} }{\mathrm{d} x} ax=a$ , $\frac{\mathrm{d} }{\mathrm{d} x} x^n=nx^{n-1}$
Note that for every derivative of a function with z, the additional term $\frac{\mathrm{d} z}{\mathrm{d} x}$ appears; this is because of the chain rule, where z=g(x), so to speak, for the function it appears in.

Using algebra to solve, we get

$\frac{\mathrm{d} z}{\mathrm{d} x}=\frac{3-z}{x+2z}$

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Example Question #31 : Implicit Differentiation

Find the normal line of the curve $\frac{x^{2}}{2}+y^{2}=4$ at the point $(1,\frac{\sqrt{14}}{2})$

Possible Answers:

$y=x\sqrt{14}-\frac{\sqrt{14}}{2}$

$y=2x\sqrt{14}-\frac{\sqrt{14}}{2}$

$y=x\sqrt{14}-{\sqrt{14}}$

$y=x\sqrt{14}+\frac{\sqrt{14}}{2}$

Correct answer:

$y=x\sqrt{14}-\frac{\sqrt{14}}{2}$

Explanation:

$\frac{x^{2}}{2}+y^{2}=4$

Use implicit differentiation to calculate the slope of the tangent line:

$\frac{\mathrm{d} }{\mathrm{d} x}(\frac{x^{2}}{2}+y^{2}=4)$

$\frac{2x}{2}+2y\frac{dy}{dx}=0$

Simplify: $x+2y\frac{dy}{dx}=0$

Subtract x from both sides of the equation: $2y\frac{dy}{dx}=-x$

Divide both sides of the equation by 2y: $\frac{dy}{dx}=\frac{-x}{2y}$

Plug in the x and y values from the point $(1,\frac{\sqrt{14}}{2})$ into the equation, to calculate the slope of the tangent line: $\frac{dy}{dx}=\frac{-1}{\frac{2\sqrt{14}}{2}}=\frac{-1}{\sqrt{14}}=\frac{-\sqrt{14}}{14}$

Take the negative reciprocal of the slope of the tangent line to calculate the slope of the normal line: $m=-\frac{1}{\frac{-\sqrt{14}}{14}}=\sqrt{14}$

Plug the slope of the normal line into point slope form: $y-\frac{\sqrt{14}}{2}=(x-1)\sqrt{14}$

Add $\frac{\sqrt{14}}{2}$ to both sides to isolate y: $y=(x-1)\sqrt{14}+\frac{\sqrt{14}}{2}$

Simplify the equation: $y=x\sqrt{14}-\sqrt{14}+\frac{\sqrt{14}}{2}$

Simplify further: $y=x\sqrt{14}-\frac{\sqrt{14}}{2}$

Solution: $y=x\sqrt{14}-\frac{\sqrt{14}}{2}$

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Example Question #1 : Finding Derivative Of A Function

What is the derivative of x^2+5x ?

Possible Answers:

x^2

$\frac{1}{2}x+5$

2x+5

$\frac{1}{3}x^3+\frac{5}{2}x^2+c$

Correct answer:

2x+5

Explanation:

To solve this problem, we can use the power rule. That means we lower the exponent of the variable by one and multiply the variable by that original exponent.

$\frac{\mathrm{d} }{\mathrm{d} x}(x^2+5x)=(2*x^{2-1})+(1*5x^{1-0})$

$\frac{\mathrm{d} }{\mathrm{d} x}(x^2+5x)=2x+5x^0$

Remember that anything to the zero power is one.

$\frac{\mathrm{d} }{\mathrm{d} x}(x^2+5x)=2x+5*1$

$\frac{\mathrm{d} }{\mathrm{d} x}(x^2+5x)=2x+5$

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AP Calculus AB : Derivatives

Study concepts, example questions & explanations for AP Calculus AB

All AP Calculus AB Resources

Example Questions

Example Question #21 : Implicit Differentiation

Example Question #22 : Implicit Differentiation

Example Question #23 : Implicit Differentiation

Example Question #21 : Implicit Differentiation

Example Question #27 : Implicit Differentiation

Example Question #81 : Applications Of Derivatives

Example Question #21 : Implicit Differentiation

Example Question #21 : Implicit Differentiation

Example Question #31 : Implicit Differentiation

Example Question #1 : Finding Derivative Of A Function

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