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Calculus 1 : Differential Functions

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Example Questions

Example Question #821 : How To Find Differential Functions

Determine if the piecewise function is differentiable:

$f(x)=\begin{Bmatrix} x^3-5x, x\leq 2\\ 7x-16, x>2 \end{Bmatrix}$

Possible Answers:

It is differentiable and continuous

It is continuous but not differentiable

It is neither continuous nor differentiable

It is differentiable but not continuous

Correct answer:

It is differentiable and continuous

Explanation:

Remember, for a function to be differentiable, it must be continuous and differentiable at all points.

Since both functions are smooth and continuous, we look at their behavior at their intersection at x=2

$f(x)=\begin{Bmatrix} x^3-5x, x\leq 2\\ 7x-16, x>2 \end{Bmatrix}$

For the first function,

f(2)=2^3-5*2= -2

For its derivative, we use the power rule:

f'(x)= 3x^2-5 , f'(2)=3*2^2-5=7

For the second function:

f(2)=7*2-16=-2

For the second function's derivative, we use the power rule:

f'(x)=7, f(2)=7

Since both the derivatives and the function values agree, this function is differentiable at all points.

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Example Question #821 : How To Find Differential Functions

Find the derivative.

3x^4-2x^2+3x

Possible Answers:

12x^3-4x+3

12x^3-4x

12x^3+4x

12x^3+4x-3

Correct answer:

12x^3-4x+3

Explanation:

Use the power rule to find the derivative.

$\frac{d}{dx}3x^4=12x^3$

$\frac{d}{dx}-2x^2=-4x$

$\frac{d}{dx}3x=3$

Thus, the derivative is 12x^3-4x+3 .

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Example Question #821 : Other Differential Functions

Find the derivative.

$\frac{8}{x^3}$

Possible Answers:

$\frac{-24}{x^4}$

$\frac{24}{x^6}$

$\frac{24}{x^4}$

$\frac{-24}{x^6}$

Correct answer:

$\frac{-24}{x^4}$

Explanation:

Use the quotient rule to find the derivative.

$\frac{d}{dx}\frac{8}{x^3}=\frac{x^3(0)-8(3x^2)}{x^6}=\frac{-24x^2}{x^6}=\frac{-24}{x^4}$

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Example Question #822 : Other Differential Functions

Find the derivative.

$\frac{x^2}{7}$

Possible Answers:

$\frac{x}{7}$

$\frac{1}{7}$

$\frac{2x^2}{7}$

$\frac{2x}{7}$

Correct answer:

$\frac{2x}{7}$

Explanation:

Use the quotient rule to find the answer.

$\frac{7(2x)-x^2(0)}{49}$

Simplify.

$\frac{14x}{49}=\frac{2x}{7}$

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Example Question #823 : Other Differential Functions

Find the derivative.

$2x^2\sin (x)$

Possible Answers:

$2x^2\cos (x)+x\sin (x)$

$2x^2\cos (x)-4x\sin (x)$

$2x^2\cos (x)-x\sin (x)$

$2x^2\cos (x)+4x\sin (x)$

Correct answer:

$2x^2\cos (x)+4x\sin (x)$

Explanation:

Use the quotient rule to find the derivative.

$2x^2\cos (x)+4x\sin (x)$

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Example Question #824 : Other Differential Functions

Find the derivative.

-3x^2+5

Possible Answers:

-6x

Correct answer:

-6x

Explanation:

Use the power rule to find the derivative.

$\frac{d}{dx}-3x^2=-6$

Recall that the derivative of a constant is zero.

Thus, the derivative is -6.

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Example Question #2046 : Calculus

Find $\frac{\mathrm{d} y}{\mathrm{d} x}$ using implicit differentiation:

x^2+5e^xy=10y^3

Possible Answers:

$\frac{\mathrm{d} y}{\mathrm{d} x}=\frac{2x+5e^x}{30y^2-5e^x}$

$\frac{\mathrm{d} y}{\mathrm{d} x}=\frac{2x+5e^xy}{30y^2-5e^xy}$

$\frac{\mathrm{d} y}{\mathrm{d} x}=\frac{2x+5e^xy}{30y^2+5e^x}$

$\frac{\mathrm{d} y}{\mathrm{d} x}=\frac{2x+5e^xy}{30y^2-5e^x}$

Correct answer:

$\frac{\mathrm{d} y}{\mathrm{d} x}=\frac{2x+5e^xy}{30y^2-5e^x}$

Explanation:

To solve for $\frac{\mathrm{dy} }{\mathrm{d} x}$ , we differentiate using normal rules, but when taking the derivative of y with respect to x, we must add the term we are solving for, $\frac{\mathrm{d} y}{\mathrm{d} x}$ .

Taking the derivative, we get

$2x+5e^xy+5e^x\frac{\mathrm{dy} }{\mathrm{d} x}=30y^2\frac{\mathrm{dy} }{\mathrm{d} x}$

using the following rules:

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

Finally, we solve for $\frac{\mathrm{d} y}{\mathrm{d} x}$ :

$\frac{\mathrm{d} y}{\mathrm{d} x}=\frac{2x+5e^xy}{30y^2-5e^x}$ .

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Example Question #831 : How To Find Differential Functions

Determine if the function is differentiable for all :

$f(t)=\begin{Bmatrix} \frac{1}{x}, x<-1\\ ln(-x-1+e), x\geq -1 \end{Bmatrix}$

Possible Answers:

The function is differentiable but not continuous for all

The function is differentiable for all

The answer cannot be determined without analysis in the complex plane

The function is not differentiable for all

Correct answer:

The function is not differentiable for all

Explanation:

When looking at differentiability of piecewise functions over all , first consider if the two functions are continuous and differentiable for all x.

$\frac{1}{x}$ is discontinuous at x=0 , but that's okay $\frac{1}{x}$ is restricted for x<-1 . However ln(-x) does not exist for $x\geq0$ . Since this is the case, we can say that this piecewise function is not differentiable for all .

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Example Question #1015 : Differential Functions

What is the second derivative of f(x)=6x^3-2x^2+9 ?

Possible Answers:

f{}''(x)=-36x-4

f{}''(x)=36x-4

f{}''(x)=4x

f{}''(x)=36x+4

f{}''(x)=36x

Correct answer:

f{}''(x)=36x-4

Explanation:

To find the second derivative, you must first find the first derivative. When taking the derivative, multiply the exponent by the coefficient in front of the x term and then subtract one from the exponent. Therefore, the first derivative is: 18x^2-4x . Then, take the derivative again to get the second derivative: f{}''(x)=36x-4 .

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Example Question #831 : Other Differential Functions

What is the derivative of f(x)=(4x^2-1)(9)

Possible Answers:

f{}'(x)=36x

f{}'(x)=72x+1

f{}'(x)=72x

f{}'(x)=2x

f{}'(x)=-72x

Correct answer:

f{}'(x)=72x

Explanation:

First, distribute the 9 to get 36x^2-9 . Then, take the derivative, remembering to multiply the exponent by the coefficient in front of the x and then subtracting one from the exponent to get an answer of f{}'(x)=72x .

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Calculus 1 : Differential Functions

Study concepts, example questions & explanations for Calculus 1

All Calculus 1 Resources

Example Questions

Example Question #821 : How To Find Differential Functions

Example Question #821 : How To Find Differential Functions

Example Question #821 : Other Differential Functions

Example Question #822 : Other Differential Functions

Example Question #823 : Other Differential Functions

Example Question #824 : Other Differential Functions

Example Question #2046 : Calculus

Example Question #831 : How To Find Differential Functions

Example Question #1015 : Differential Functions

Example Question #831 : Other Differential Functions

All Calculus 1 Resources

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