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AP Calculus AB : Applications of Derivatives

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Example Question #8 : Implicit Differentiation

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

$\frac{y^3}{3}+x^3=ye^{2x}$

Possible Answers:

$\frac{2ye^{2x}-3x^2}{y^2-e^{2x}}$

$\frac{2ye^{2x}+3x^2}{y^2-e^{2x}}$

$\frac{2ye^{2x}-y^2}{3x^2-e^{2x}}$

$\frac{y^2-e^{2x}}{2ye^{2x}-3x^2}$

$\frac{2ye^{2x}-3x^2}{y^2+e^{2x}}$

Correct answer:

$\frac{2ye^{2x}-3x^2}{y^2-e^{2x}}$

Explanation:

To find the derivative of y with respect to x, we must take the differentiate both sides of the equation with respect to x:

$y^2 \frac{\mathrm{d} y}{\mathrm{d} x}+3x^2=2ye^{2x}+e^{2x}\frac{\mathrm{d} y}{\mathrm{d} x}$

The following rules were used:

$\frac{\mathrm{d} }{\mathrm{d} x} f(g(x))=f'(g(x))\cdot g'(x)$ , $\frac{\mathrm{d} }{\mathrm{d} x} x^n=nx^{n-1}$ , $\frac{\mathrm{d} }{\mathrm{d} x} f(x)g(x)=f'(x)g(x)+f(x)g'(x)$ , $\frac{\mathrm{d} }{\mathrm{d} x} e^x=e^x$ , $\frac{\mathrm{d} }{\mathrm{d} x} ax=a$

Note that the chain rule was used everywhere we took the derivative of a function containing y, as well as in the exponential function. The product rule was used because both x and y are both functions of x.

Using algebra to solve, we get

$\frac{\mathrm{d} y}{\mathrm{d} x}=\frac{2ye^{2x}-3x^2}{y^2-e^{2x}}$

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Example Question #1 : Implicit Differentiation

Find $\frac{\mathrm{d} \alpha}{\mathrm{d} x}$ , where $\alpha$ is a function of x:

$\alpha^2x^2+2\alpha x=\alpha$

Possible Answers:

$\frac{2\alpha^2 x+2 \alpha}{1-2\alpha x^2 -2x}$

None of the other answers

$\frac{2\alpha^2 x+2 \alpha}{1+2\alpha x^2+2x}$

$\frac{2\alpha x^2+2 \alpha}{1-2\alpha^2 x -2x}$

Correct answer:

$\frac{2\alpha^2 x+2 \alpha}{1-2\alpha x^2 -2x}$

Explanation:

To find the derivative of $\alpha$ with respect to x, we must take the differentiate both sides of the equation with respect to x:

$2\alpha x^2 \frac{\mathrm{d} \alpha}{\mathrm{d} x}+2\alpha^2x+2x \frac{\mathrm{d} \alpha}{\mathrm{d} x}+2x=\frac{\mathrm{d} \alpha}{\mathrm{d} x}$

The following derivative rules were used:

$\frac{\mathrm{d} }{\mathrm{d} x} f(g(x))=f'(g(x))\cdot g'(x)$ , $\frac{\mathrm{d} }{\mathrm{d} x} f(x)g(x)=f'(x)g(x)+f(x)g'(x)$ , $\frac{\mathrm{d} }{\mathrm{d} x} x^n=nx^{n-1}$ , $\frac{\mathrm{d} }{\mathrm{d} x} ax=a$

Note that the chain rule was used for the derivative of any function containing $\alpha$ , whose derivative with respect to x we want to solve for.

Solving, we get

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

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Example Question #10 : Implicit Differentiation

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

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

Possible Answers:

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

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

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

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

Correct answer:

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

Explanation:

To solve this using implicit differentiation, we must always treat y as a function of x, and therefore when we differentiate y with respect to x, we denote it as

Step by step, we get the following:

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

This resulted from the product rule and chain rule

The next steps are:

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

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

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

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

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Example Question #11 : Implicit Differentiation

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

$y=y\cos(x^3)$

Possible Answers:

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

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

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

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

Correct answer:

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

Explanation:

To find the derivative of the function, we use implicit differentiation, where we always treat y as a function of x, and denoting any derivative of y with respect to x as

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

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

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

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

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Example Question #11 : Implicit Differentiation

Find $\frac{\mathrm{d} y}{\mathrm{d} x}$ from the following equation:

$e^{2y}+x^2y^2=y$

Possible Answers:

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

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

$\frac{-2xy^2-2x^2y}{2e^{2y}-1}$

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

Correct answer:

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

Explanation:

To find the derivative of y with respect to x, we must take the derivative with respect to x of both sides of the equation:

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

The derivatives were found using the following rules:

$\frac{\mathrm{d} }{\mathrm{d} x} f(g(x))=f'(g(x))\cdot g'(x)$ , $\frac{\mathrm{d} }{\mathrm{d} x} e^u=e^u\frac{\mathrm{d} u}{\mathrm{d} x}$ , $\frac{\mathrm{d} }{\mathrm{d} x} x^n=nx^{n-1}$

Notice the chain rule was used for every function containing y, because y is a function of x and its whose derivative we are interested in isolating.

Using algebra to solve, our final answer is

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

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Example Question #13 : Implicit Differentiation

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

$y^4=\sin(2x)$

Possible Answers:

$\frac{\cos(2x)}{2y^3}$

$2\cos(2x)$

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

$\frac{4\cos(2x)}{y^3}$

Correct answer:

$\frac{\cos(2x)}{2y^3}$

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

$4y^3 \frac{\mathrm{d} y}{\mathrm{d} x}=2\cos(2x)$

The derivatives were found using the following rules:

$\frac{\mathrm{d} }{\mathrm{d} x} f(g(x))=f'(g(x))\cdot g'(x)$ , $\frac{\mathrm{d} }{\mathrm{d} x} x^n=nx^{n-1}$ , $\frac{\mathrm{d} }{\mathrm{d} x} \sin(x)=\cos(x)$ , $\frac{\mathrm{d} }{\mathrm{d} x} ax=a$

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.

Solving for $\frac{\mathrm{d} y}{\mathrm{d} x}$ , we get

$\frac{\mathrm{d} y}{\mathrm{d} x}=\frac{\cos(2x)}{2y^3}$

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Example Question #14 : Implicit Differentiation

Evaluate $\frac{\mathrm{d}y }{\mathrm{d} x}$ at the point (1, 4) for the following equation:

xy + 3x^2 - 2y^2 = -25 .

Possible Answers:

$\frac{2}{3}$

$\frac{3}{2}$

$\frac{2}{15}$

$-\frac{2}{3}$

$\frac{15}{2}$

Correct answer:

$\frac{2}{3}$

Explanation:

Using implicit differentiation, taking the derivative of the given equation yields

$y + \frac{\mathrm{d} y}{\mathrm{d} x}x + 6x -4y\frac{\mathrm{d} y}{\mathrm{d} x} = 0$

Getting all the $\frac{\mathrm{d} y}{\mathrm{d} x}$ 's to their own side, we have

$x\frac{\mathrm{d} y}{\mathrm{d} x} - 4y\frac{\mathrm{d} y}{\mathrm{d} x} = -6x - y$

Factoring,

$\frac{\mathrm{d} y}{\mathrm{d} x}(x - 4y) = -6x - y$

And dividing

$\frac{\mathrm{d}y }{\mathrm{d} x} = \frac{-6x - y}{x - 4y}$

Plugging in our point, (1, 4) , we have

$\frac{\mathrm{d} y}{\mathrm{d} x} = \frac{-6 - 4}{1 - 16} = \frac{-10}{-15} =\frac{2}{3}$

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Example Question #11 : Implicit Differentiation

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

$y^3+\sec(y)=3x^2$

Possible Answers:

$\frac{6x+3y^2}{\sec(y)\tan(y)}$

$\frac{6x}{3y^2+\sec(y)\tan(y)}$

$\frac{6x}{3y^2+\sec^2(y)}$

$\frac{3y^2+\sec(y)\tan(y)}{6x}$

Correct answer:

$\frac{6x}{3y^2+\sec(y)\tan(y)}$

Explanation:

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

Taking the derivative with respect to x of both sides of the equation, we get

$3y^2 \frac{\mathrm{d} y}{\mathrm{d} x}+\sec(y)\tan(y)\frac{\mathrm{d} y}{\mathrm{d} x}=6x$

The derivatives were found using the following rules:

$\frac{\mathrm{d} }{\mathrm{d} x} f(g(x))=f'(g(x))\cdot g'(x)$ , $\frac{\mathrm{d} }{\mathrm{d} x} \sec(x)=\sec(x)\tan(x)$ , $\frac{\mathrm{d} }{\mathrm{d} x} x^n=nx^{n-1}$

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

$\frac{\mathrm{d} y}{\mathrm{d} x}=\frac{6x}{3y^2+\sec(y)\tan(y)}$

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Example Question #601 : Derivatives

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

ye^y+x=2xy

Possible Answers:

$\frac{2y-1+e^y}{-ye^y-2x}$

$\frac{2y-1}{e^y-ye^y-2x}$

$\frac{2y-1+2x}{e^y-ye^y}$

$\frac{2y-1}{e^y-2x}$

Correct answer:

$\frac{2y-1}{e^y-ye^y-2x}$

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

$e^y \frac{\mathrm{d} y}{\mathrm{d} x}+ye^y\frac{\mathrm{d} y}{\mathrm{d} x}+1=2y+2x\frac{\mathrm{d} y}{\mathrm{d} x}$

which was found using the following rules:

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

$\frac{\mathrm{d} y}{\mathrm{d} x}=\frac{2y-1}{e^y-ye^y-2x}$

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Example Question #61 : Applications Of Derivatives

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

$y=6\sin(xy)+e^y$

Possible Answers:

$y'=\frac{6y\cos(xy)}{1-6x\cos(xy)-e^y}$

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

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

$y'=\frac{6y\cos(xy)}{1-6\cos(xy)-ye^y}$

Correct answer:

$y'=\frac{6y\cos(xy)}{1-6x\cos(xy)-e^y}$

Explanation:

To find the derivative of the function, we use implicit differentiation, which is an application of the chain rule. We use this because y=y(x) , and any derivative with respect to is $\frac{\mathrm{d} y}{\mathrm{d} x}$ (or ).

First, we use the chain rule combined with the product rule in taking the derivative of y

$y'=6\cos(xy)(xy'+y)+y'e^y$

Then we expand in order to isolate the terms with

$y'=6xy'\cos(xy)+6y\cos(xy)+y'e^y$

$y'-6xy'\cos(xy)-y'e^y=6y\cos(xy)$

Then we factor out a

$y'(1-6x\cos(xy)-e^y)=6y\cos(xy)$

$y'=\frac{6y\cos(xy)}{1-6x\cos(xy)-e^y}$

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AP Calculus AB : Applications of Derivatives

Study concepts, example questions & explanations for AP Calculus AB

All AP Calculus AB Resources

Example Questions

Example Question #8 : Implicit Differentiation

Example Question #1 : Implicit Differentiation

Example Question #10 : Implicit Differentiation

Example Question #11 : Implicit Differentiation

Example Question #11 : Implicit Differentiation

Example Question #13 : Implicit Differentiation

Example Question #14 : Implicit Differentiation

Example Question #11 : Implicit Differentiation

Example Question #601 : Derivatives

Example Question #61 : Applications Of Derivatives

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