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Calculus 1 : Other Differential Functions

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Example Questions

Example Question #51 : Other Differential Functions

Differentiate the function using known derivatives and applying the product, quotient, and chain rules.

$f(x)=(x^{3}+4x^{2}+3)^{2}$

Possible Answers:

$f'(x)=2*(3x^2 + 8x)^{2}$

$f'(x)=(x^{3}+4x^{2}+3)^{2}(3x^2 + 8x)$

$f'(x)=2x^{3}+8x^{2}+6$

f'(x)=2*(3x^2 + 8x)(6x+8)

$f'(x)=2*(x^{3}+4x^{2}+3)(3x^2 + 8x)$

Correct answer:

$f'(x)=2*(x^{3}+4x^{2}+3)(3x^2 + 8x)$

Explanation:

$f(x)=(x^{3}+4x^{2}+3)^{2}$

We evaluate this derivative using the chain rule:

$if \ f(x)=g(h(x))$ ,

$then\ f'(x)=g'(h(x))*h'(x)$ .

The outside function is:

$g(x) = x^2 \ so \ g'(x) = 2x$

The inside function is:

$h(x)=x^{3}+4x^{2}+3 \ so \ h'(x) = 3x^2 + 8x$

Therefore,

$f'(x)=2*(x^{3}+4x^{2}+3)(3x^2 + 8x)$ , which is our final answer.

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Example Question #231 : Functions

Solve the indefinite integral. If you cannot evaluate directly, use u-substitution.

$\int tan(x)dx$

Possible Answers:

-ln(cosx)+c

ln(sinx)+c

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

$\frac{sinx}{cosx} + c$

$sec^{2}x + c$

Correct answer:

-ln(cosx)+c

Explanation:

We use the trig identity $tanx = \frac{sinx}{cosx}$ .

$\int tan(x)dx = \int \frac{sin(x)}{cos(x)}dx$

$Let \ u = cos(x), \ du=-sin(x)dx.$

$\int \frac{sin(x)}{cos(x)}dx = -\int \frac{du}{u}=-ln(u) +c= -ln(cosx)+c$ , which is our final answer.

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Example Question #1269 : Calculus

Solve the indefinite integral. If you cannot evaluate directly, use u-substitution.

$\int xe^{x^2}dx$

Possible Answers:

$\frac{1}{2}e^{x^{2}}$ + c

$2x^2e^{x^{2}}$ + c

$2e^{x^{2}}$ + c

$e^{2x}$ + c

$\frac{1}{2}e^{2x}$ + c

Correct answer:

$\frac{1}{2}e^{x^{2}}$ + c

Explanation:

$\int xe^{x^2}dx$

$Let \ u = x^2, \ du=2xdx$ .

$\int xe^{x^2}dx = \frac{1}{2}\int 2xe^{x^2}dx =\frac{1}{2}\int e^udu = \frac{1}{2}e^u = \frac{1}{2}e^{x^{2}}$ , which is our final answer.

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Example Question #241 : Functions

Solve the indefinite integral. If you cannot evaluate directly, use u-substitution.

$\int \frac{ln(x)}{x}dx$

Possible Answers:

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

$\frac{1}{ln(x)} +c$

$-\frac{ln(x) }{x^2}+c$

ln(ln(x))+c

$\frac{1}{2}ln(x)^2 +c$

Correct answer:

$\frac{1}{2}ln(x)^2 +c$

Explanation:

$\int \frac{ln(x)}{x}dx$

$Let \ u = ln(x), \ du=\frac{1}{x}dx$ .

$\int \frac{ln(x)}{x}dx = \int udu = \frac{1}{2}u^2 + c = \frac{1}{2}ln(x)^2 +c$ , which is our final answer.

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Example Question #1271 : Calculus

Solve the indefinite integral. If you cannot evaluate directly, use u-substitution.

$\int \frac{e^{x^{\frac{1}{2}}}}{x^{\frac{1}{2}}}dx$

Possible Answers:

$e^{x^{\frac{3}{2}}} + c$

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

$\frac{1}{2}e^{x^{\frac{1}{2}}} + c$

$2e^{x^{\frac{1}{2}}} + c$

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

Correct answer:

$2e^{x^{\frac{1}{2}}} + c$

Explanation:

$\int \frac{e^{x^{\frac{1}{2}}}}{x^{\frac{1}{2}}}dx$

We rewrite the denominator as a negative exponenet in the numerator to make the u-substitution easier to see:

$\int \frac{e^{x^{\frac{1}{2}}}}{x^{\frac{1}{2}}}dx = \int e^{x^{\frac{1}{2}}}}{x^{- \frac{1}{2}}dx$

$Let\: u=x^{1/2}, du=\frac{1}{2}x^{-1/2}dx.$

$\int e^{x^{\frac{1}{2}}}}{x^{- \frac{1}{2}}dx = 2\int \frac{1}{2}e^{x^{\frac{1}{2}}}}{x^{- \frac{1}{2}}dx = 2 \int e^udu = 2e^u + c = 2e^{x^{\frac{1}{2}}} + c$ , which is our final answer.

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Example Question #242 : Functions

Solve the indefinite integral. If you cannot evaluate directly, use u-substitution.

$\int 2sinx*cosxdx$

Possible Answers:

sin^2x +c

ln(sinx)+c

tanx +c

-2sinx*cosx +c

2sinx*cosx +c

Correct answer:

sin^2x +c

Explanation:

$\int 2sinx*cosxdx$

$Let\: u=sinx,\: du=cosxdx.$

$\int 2sinx*cosxdx = \int 2udu = u^2 +c = sin^2x +c$ , which is our final answer.

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Example Question #241 : Differential Functions

Solve the indefinite integral. If you cannot evaluate directly, use u-substitution.

$\int \frac{e^{(1+lnx)}}{x}dx$

Possible Answers:

$\frac{-e^{(1+lnx)}}{x^2}+c$

$\frac{x}{1+lnx}+c$

$e^{(1+lnx)}+c$

$e^{(x+xlnx)}+c$

1+lnx+c

Correct answer:

$e^{(1+lnx)}+c$

Explanation:

$\int \frac{e^{(1+lnx)}}{x}dx$

$Let \ u = 1+ln(x)), \ du=\frac{1}{x}dx.$

$\int \frac{e^{(1+lnx)}}{x}dx = \int e^udu= e^u +c = e^{(1+lnx)}+c$ , which is our final answer.

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Example Question #51 : Other Differential Functions

Find the derivative of x^2 - sin(x) .

Possible Answers:

2x+sin(x)

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

2x+cos(x)

2x - cos(x)

2x - sin(x)

Correct answer:

2x - cos(x)

Explanation:

The derivative of the difference of two functions is the difference of the derivative of the two functions:
$\frac{dy}{dx} = 2x - \frac{d}{dx}(sin(x))$
= 2x - cos(x)

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Example Question #52 : Other Differential Functions

Find the derivative of $\frac{1}{(3x^2+5)^4}$ .

Possible Answers:

$\frac{-4x}{(3x^2+5)^5}$

$\frac{-24x}{(3x^2+5)^4}$

$\frac{24x}{(3x^2+5)^5}$

$\frac{-24x}{(3x^2+5)^5}$

Correct answer:

$\frac{-24x}{(3x^2+5)^5}$

Explanation:

We can write the function as

$\frac{1}{(3x^2+5)^4}=(3x^2+5)^{-4}}$ .

Let $u(x)=3x^2+5\Rightarrow \frac{du}{dx}=6x$ .

We then have

$\frac{dy}{dx}=\frac{dy}{du}*\frac{du}{dx}=-4(3x^2+5)^{-5}*(6x)=\frac{-24x}{(3x^2+5)^5}$ .

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Example Question #53 : Other Differential Functions

Differentiate the function:

$f(x)=e^{x^{2}}$

Possible Answers:

$f'(x)= e^{2x}$

$f'(x)= 2x*e^{x^{2}}$

$f'(x)= x^{2}*e^{x^{2}}$

$f'(x)= e^{x^{2}}$

$f'(x)= 2x*e^{2x}$

Correct answer:

$f'(x)= 2x*e^{x^{2}}$

Explanation:

$f(x)=e^{x^{2}}$

Using the chain rule, $if \ f(x)=g(h(x))$ , $then\ f'(x)=g'(h(x))*h'(x)$ ,

we observe the following:

$g(x)= e^x \ so\ g'(x)= e^x$ .

$h(x) = x^2 \ so \ h'(x)=2x$ .

$f'(x)= 2x*e^{x^{2}}$

$f'(x)=g'(h(x))*h'(x) = e^{x^{2}}*2x = 2x*e^{x^{2}}$ , which is our final answer.

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Calculus 1 : Other Differential Functions

Study concepts, example questions & explanations for Calculus 1

All Calculus 1 Resources

Example Questions

Example Question #51 : Other Differential Functions

Example Question #231 : Functions

Example Question #1269 : Calculus

Example Question #241 : Functions

Example Question #1271 : Calculus

Example Question #242 : Functions

Example Question #241 : Differential Functions

Example Question #51 : Other Differential Functions

Example Question #52 : Other Differential Functions

Example Question #53 : Other Differential Functions

All Calculus 1 Resources

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