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Calculus 1 : How to find differential functions

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Example Questions

Example Question #821 : How To Find Differential Functions

Find the first derivative of y=sin^3(5x+4)

Possible Answers:

y'=3sin^2(5x+4)

y'=15 sin^2(5x+4) cos(5x+4)

y'=15cos^2(5x+4)

y'=5cos^3(5x+4)

y'=15sin^2(5x+4)

Correct answer:

y'=15 sin^2(5x+4) cos(5x+4)

Explanation:

We need to use the chain rule TWICE, which says

$\frac{d}{dx}(f(g(h(x))))=f'(g(h(x)))\cdot g'(h(x))\cdot h'(x)$

Here,

f(x)=x^3, g(x)=sin(x), h(x)=5x+4

Applying the rule to differentiate,

$y'=3\cdot sin^2(5x+4) \cdot cos(5x+4) \cdot 5$

Simplifying,

y'=15 sin^2(5x+4) cos(5x+4)

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Example Question #1006 : Differential Functions

Find the first derivative of y=x^3-x^2sin(x)

Possible Answers:

y'=x(3x+xcos(x)-2sin(x))

y'=3x^2-2xsin(x)

y'=3x^2-2xcos(x)

y'=3x^2-2x-cos(x)

y'=3x^2-2xsin(x)cos(x)

Correct answer:

y'=x(3x+xcos(x)-2sin(x))

Explanation:

We will need to use the product rule to differentiate the second term, which says

$\frac{d}{dx}\left [f(x)\cdot g(x) \right ]=f(x)g'(x)+f'(x)g(x)$

Applying to differentiate y,

$y'=3x^2-\left [ x^2(sin(x))'+(x^2)'sin(x)\right ]$

$y'=3x^2-\left [ x^2(-cos(x))+2xsin(x)\right ]$

y'=3x^2+x^2cos(x)-2xsin(x)

Factoring to simplify, gives

y'=x(3x+xcos(x)-2sin(x))

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Example Question #2032 : Calculus

Find the derivative of y using implicit differentiation for the following function:

x=2xy+3y^2

Possible Answers:

$\frac{\mathrm{dy} }{\mathrm{d} x}=8y$

$\frac{\mathrm{dy} }{\mathrm{d} x}=\frac{1-2y}{2x+6y}$

$\frac{\mathrm{dy} }{\mathrm{d} x}=2x+6y$

$\frac{\mathrm{dy} }{\mathrm{d} x}=\frac{1+2y}{2x+6y}$

Correct answer:

$\frac{\mathrm{dy} }{\mathrm{d} x}=\frac{1-2y}{2x+6y}$

Explanation:

To solve for y' - the derivative of y - we must use implicit differentiation. All of the normal differentiation rules apply, but when we take the derivative of y with respect to x we must always include $\frac{\mathrm{dy} }{\mathrm{d} x}$ .

For the function given, when we take the derivative of both sides of the equation with respect to x, we get

$1=2y+2x\frac{\mathrm{dy} }{\mathrm{d} x}+6y\frac{\mathrm{dy} }{\mathrm{d} x}$

using the following rules:

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

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

$1-2y=\frac{\mathrm{dy} }{\mathrm{d} x}(2x+6y)$

$\frac{\mathrm{dy} }{\mathrm{d} x}=\frac{1-2y}{2x+6y}$ .

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Example Question #2031 : Calculus

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 neither continuous nor differentiable

It is differentiable but not continuous

It is continuous but not differentiable

It is differentiable and 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 #822 : How To Find Differential Functions

Find the derivative.

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

Possible Answers:

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

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

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

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

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 #823 : How To Find Differential Functions

Find the derivative.

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

Possible Answers:

$\frac{x}{7}$

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

$\frac{2x}{7}$

$\frac{1}{7}$

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 #824 : How To Find Differential Functions

Find the derivative.

$2x^2\sin (x)$

Possible Answers:

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

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

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

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

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 #825 : How To Find 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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Calculus 1 : How to find 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 #1006 : Differential Functions

Example Question #2032 : Calculus

Example Question #2031 : Calculus

Example Question #821 : How To Find Differential Functions

Example Question #822 : How To Find Differential Functions

Example Question #823 : How To Find Differential Functions

Example Question #824 : How To Find Differential Functions

Example Question #825 : How To Find Differential Functions

Example Question #2046 : Calculus

All Calculus 1 Resources

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