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Calculus 1 : Differential Functions

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Example Questions

Example Question #131 : Other Differential Functions

Differentiate the function:

$f(x)=3-\frac{2}{3}x$

Possible Answers:

$f'(x)=-\frac{2}{3}$

$f'(x)=-\frac{2}{3}x^-^1$

f'(x)=3-x

$f'(x)=1-\frac{2}{3}x$

Correct answer:

$f'(x)=-\frac{2}{3}$

Explanation:

When the function has a sum or difference of terms, take the derivate with respect to the variable (in the case x) of each term:

$f(x)=3-\frac{2}{3}x$

Using the power rule which states,

$f(x)=x^n \rightarrow f'(x)=nx^{n-1}$

we find our derivative to be,

$f'(x)=\frac{d}{dx}(3)-\frac{d}{dx}\left(\frac{2}{3}x\right)=0-1\cdot\frac{2}{3}\cdot x^1^-^1=-\frac{2}{3}$ .

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Example Question #321 : Differential Functions

Differentiate the equation:

$f(x)=\pi^2$

Possible Answers:

$f'(x)=2\pi$

f'(x)=0

$f'(x)=2x\pi$

{}f'(x)=2x

Correct answer:

f'(x)=0

Explanation:

Becuase $\pi^2$ is a constant, every derivative of will be 0.

The general rule for differentiating a constant , is as follows.

$f(x)=c\rightarrow f'(x)=0$

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Example Question #322 : Differential Functions

Differentiate the function:

$f(t)=\frac{3}{4}t^1^2$

Possible Answers:

f'(t)=9x^1^1

f'(t)=9x^1^2

$f'(t)=\frac{3}{4}^8$

$f'(t)=\frac{3}{4}x^1^1$

Correct answer:

f'(t)=9x^1^1

Explanation:

Use the power rule: nx^n^-^1 and multiply the exponent by the coefficient then decrease the exponent by one to find the derivative of the function.

$f(t)=\frac{3}{4}t^1^2$

where n=12

$f'(t)=12\cdot \frac{3}{4}x^1^2^-^1=9x^1^1$

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Example Question #322 : Differential Functions

Differentiate the function:

f(a)=a^4-a+7

Possible Answers:

f'(a)=4a^4-a

f'(a)=4a+6

f'(a)=4a^3-1

f'(a)=a^3-1

Correct answer:

f'(a)=4a^3-1

Explanation:

When the function consists of the sum or differenct of terms, take the derivative of each term with respect to the variable (a).

Using the power rule which states,

$f(x)=x^n \rightarrow f'(x)=nx^{n-1}$

we find our derivative to be,

$f'(a)=\frac{d}{da}(a^4)-\frac{d}{da}(a)+\frac{d}{da}(7)=4\cdot a^4^-^3-1\cdota^1^-^1+0=4a^3-1$ .

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Example Question #321 : Functions

Differentiate the function:

f(x)=1.5x^2-3.4x+7

Possible Answers:

f'(x)=3x-3.4

f'(x)=3x

f'(x)=1.5x+3.4

f'(x)=5.1

Correct answer:

f'(x)=3x-3.4

Explanation:

When a function has a sum or difference of terms, take the derivative of each term with respect to x.

To take the derivative we will need to use the power rule which states,

$f(x)=x^n \rightarrow f'(x)=nx^{n-1}$

Applying this rule term by term, we find the derivative as follows.

$f'(x)=\frac{d}{dx}\left(1.5x^2\right)-\frac{d}{dx}(3.4x)+\frac{d}{dx}(7)$

$f'(x)=2\cdot 1.5x^2^-^1-1\cdot 3.4x^1^-^1+0$

f'(x)=3x-3.4

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Example Question #135 : Other Differential Functions

Differentiate the function:

f(b)=sin(b)-9cos(b)

Possible Answers:

f'(b)=cos(b)-9

f'(b)=-cos(b)-9sin(b)

f'(b)=cos(b)+9sin(b)

f'(b)=-9sin(b)+cos(b)

Correct answer:

f'(b)=cos(b)+9sin(b)

Explanation:

Take the derivative of each term with respect to b.

To take the derivative we will need to use the power rule which states,

$f(x)=x^n \rightarrow f'(x)=nx^{n-1}$ .

Also recall that the derivative of sin(x) is cos(x) .

Applying these rules we find the derivative to be:

$f'(b)=\frac{d}{dx}(sin(b))-\frac{d}{dx}(9cos(b))=cos(b)-9(-sin(b))=cos(b)+9sin(b)$ .

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Example Question #323 : Differential Functions

Differentiate the function:

$f(x)=4x^2-\cos(x)$

Possible Answers:

$f'(x)=4x-\sin(x)$

$f'(x)=8x+\sin(x)$

f'(x)=8x^2+sin(x)

f'(x)=4x^2-sin(x)

Correct answer:

$f'(x)=8x+\sin(x)$

Explanation:

When a function has a sum or difference of terms, take the derivative of each term with repspect to x.

To take the derivative we will need to use the power rule which states,

$f(x)=x^n \rightarrow f'(x)=nx^{n-1}$ .

Also recall that the derivative of cos(x) is -sin(x) .

Applying these rules, we find the derivative as follows.

$f'(x)=\frac{d}{dx}(4x^2)-\frac{d}{dx}(cos(x))=2\cdot 4x^2^-^1-(-sin(x))=8x+sin(x)$

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Example Question #324 : Differential Functions

Differentiate the function:

f(z)=az^-^7

Possible Answers:

f'(z)=-7az^-^8

f'(z)=-7az^-^6

f'(z)=-7z^-^8

f'(z)=7az^6

Correct answer:

f'(z)=-7az^-^8

Explanation:

To differentiate this problem we will need to use the power rule.

The power rule is, nx^n^-^1 where n is the exponent.

Thus our derivative is,

$f'(z)=\frac{d}{dz}(az^-^7)=-7\cdot az^-^7^-^1=-7az^-^8$ .

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Example Question #325 : Differential Functions

Differentiate the function:

$f(x)=-12x^\frac{-3}{4}$

Possible Answers:

$f'(x)=12x^\frac{-3}{4}$

$f'(x)=-9x^\frac{1}{4}$

$f'(x)=9x^\frac{-7}{4}$

$f'(x)=9x^\frac{1}{4}$

Correct answer:

$f'(x)=9x^\frac{-7}{4}$

Explanation:

To find the derivative of this function we will need to use the power rule.

The power rule is nx^n^-^1 where n is the exponent.

Therefore, our derivative is

$f'(x)=\left(-\frac{3}{4}\right)\cdot (-12)x^\frac{-3}{4}^-^{1}=9x^{\frac{-7}{4}}$ .

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Example Question #323 : Functions

Differentiate the function:

$f(x)=\sqrt{x}-x$

Possible Answers:

$f'(x)=-\frac{1}{2}x-1$

$f'(x)=\frac{1}{2}x^{-\frac{1}{2}}-1$

$f'(x)=\sqrt{x}-1$

$f'(x)=-\frac{1}{2}x^{\frac{1}{2}}+1$

Correct answer:

$f'(x)=\frac{1}{2}x^{-\frac{1}{2}}-1$

Explanation:

Roots of numbers are fractional exponents so express $\sqrt{x}$ as $x^{\frac{1}{2}}$ and differentiate each term with respect to x using the power rule:

$f(x)=x^n \rightarrow f'(x)=nx^{n-1}$

$f'(x)=\frac{d}{dx}(x^{\frac{1}{2}})-\frac{d}{dx}(x)=\frac{1}{2}x^{\frac{1}{2}-1}-1=\frac{1}{2}x^{-\frac{1}{2}}-1$

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Calculus 1 : Differential Functions

Study concepts, example questions & explanations for Calculus 1

All Calculus 1 Resources

Example Questions

Example Question #131 : Other Differential Functions

Example Question #321 : Differential Functions

Example Question #322 : Differential Functions

Example Question #322 : Differential Functions

Example Question #321 : Functions

Example Question #135 : Other Differential Functions

Example Question #323 : Differential Functions

Example Question #324 : Differential Functions

Example Question #325 : Differential Functions

Example Question #323 : Functions

All Calculus 1 Resources

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