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Calculus 1 : Differential Functions

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Example Questions

Example Question #281 : Differential Functions

Find the derivative of

$f(x) =8\sin(4x+3)$

Possible Answers:

$f'(x) = -32\sin(4x+3)$

$f'(x) = 32\sin(4x+3)$

$f'(x) = 32\cos(4x+3)$

$f'(x) = 8\cos(4x+3)$

Correct answer:

$f'(x) = 32\cos(4x+3)$

Explanation:

To find the derivative of the function we must use the Chain Rule

$f(x) = h(g(x)) \rightarrow f'(x) = h'(g(x))\cdot g'(x)$

Applying this to the function we are given gives,

$f'(x) = 8\cos(4x+3)(4) = 32\cos(4x+3)$

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Example Question #282 : Differential Functions

Find the first derivative of the function

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

Possible Answers:

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

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

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

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

Correct answer:

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

Explanation:

To find the derivative of this function we can use the Product Rule

f'(x) = h'(x)g(x)+h(x)g'(x)

Applying this to the function we get

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

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Example Question #282 : Differential Functions

Differentiate the following function

$f(x) = \frac{\cos(x)}{x^2}$

Possible Answers:

$f'(x) = \frac{x^2\sin(x)+2x\cos(x)}{x^4}$

None of these answers are correct.

$f'(x) = \frac{x^2\sin(x)+2x\cos(x)}{x^2}$

$f'(x) = \frac{-x^2\sin(x)-2x\cos(x)}{x^4}$

Correct answer:

$f'(x) = \frac{-x^2\sin(x)-2x\cos(x)}{x^4}$

Explanation:

To differentiate the function that we are given we must use the Quotient Rule

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

Applying this to the function we are given gives,

$f'(x) = \frac{-\sin(x)x^2-2x\cos(x)}{x^4}$

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Example Question #96 : Other Differential Functions

Find the derivative of

$f(x) = \frac{\sin^2x}{x}$

Possible Answers:

$\small f'(x) = \frac{2\sin(x)\cos(x)+\sin^2(x)\cos(x)}{x^2}$

$\small \small f'(x) = \frac{2x\sin(x)\cos(x)-\sin^2x}{x^2}$

None of these answers are correct.

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

Correct answer:

$\small \small f'(x) = \frac{2x\sin(x)\cos(x)-\sin^2x}{x^2}$

Explanation:

To differentiate this function we must use the Chain Rule and the Quotient Rule

$\small f(x) = h(g(x)) \rightarrow f'(x) = h'(g(x))\cdot g'(x)$

$\small f'(x) = \frac{h'(x)g(x) - h(x)g'(x)}{(g(x))^2}$

Applying these to the function we are given gives us,

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

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Example Question #96 : How To Find Differential Functions

Find the first derivative of the function

$\small f(x) = \frac{\sin(x)}{\cos^2(x)}$

Possible Answers:

$\small \small \small f'(x) = \frac{\cos^3x+2\sin^2x\cos(x)}{\cos^4x}$

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

$\small \small \small f'(x) = \frac{\cos^2x-2\sin^2x\cos(x)}{\cos^4x}$

None of these answers are correct.

Correct answer:

$\small \small \small f'(x) = \frac{\cos^3x+2\sin^2x\cos(x)}{\cos^4x}$

Explanation:

To differentiate this function we must use the Quotient Rule where

$\small f'(x) = \frac{h'(x)g(x)-h(x)g'(x)}{(g(x))^2}$

Using $\small \small h(x) = \sin(x)$ and $\small g(x) = \cos^2(x)$ with the Quotient Rule gives,

$\small f'(x) = \frac{\cos(x)\cos^2(x)-2\cos(x)(-\sin(x))\sin(x)}{\cos^4(x)} \rightarrow$

$\small f'(x) = \frac{\cos^3x+2\sin^2x\cos(x)}{\cos^4x}$

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Example Question #286 : Differential Functions

Differentiate

$\small f(x) = \sin(\cos^2(x^4))$

Possible Answers:

$\small \small f'(x) = 4x^3\cos(\cos^2(x^4))\cos(x^4)\sin(x^4)$

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

$\small \small \small f'(x) = -8x^3\cos(\cos^2(x^4))\cos(x^4)\sin(x^4)$

$\small \small f'(x) = -4x^3\cos(\cos^2(x^4))\cos(x^4)\sin(x^4)$

Correct answer:

$\small \small \small f'(x) = -8x^3\cos(\cos^2(x^4))\cos(x^4)\sin(x^4)$

Explanation:

To differentiate this function we must use the Chain Rule.

$\small \small f(x) = a(b(c(d(x)))) \rightarrow \\ f'(x) = a'(b(c(d(x))))\cdot b'(c(d(x)))\cdot c'(d(x))\cdot d'(x)$

Applying this to the function we obtain,

$\small \small \small f'(x) = \cos(\cos^2(x^4))2\cos(x^4)(-\sin(x^4))(4x^3) = -8x^3\cos(\cos^2(x^4))\cos(x^4)\sin(x^4)$

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Example Question #96 : How To Find Differential Functions

Differentiate the polynomial.

$7x{^{2}}+5$

Possible Answers:

14x

$7x^{2}$

Correct answer:

14x

Explanation:

Using the power rule, we can differentiate our first term reducing the power by one and multiplying our term by the original power. $7x^{2}$ , will thus become 14x . The second term is a constant value, so according to the power rule this term will become .

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Example Question #101 : How To Find Differential Functions

Differentiate the trigonometric function.

$\cos(3x)$

Possible Answers:

$-3\sin(x)$

$-\cos(3x)$

$3\sin(3x)$

$-3\cos(3x)$

$\sin(3x)$

Correct answer:

$-3\sin(x)$

Explanation:

We can use the chain rule to differentiate, which states we will need to multiply the derivative of the outside function by the derivative of the inside function. We find the derivative of the inside function, , to be . The derivative of the outside function $\cos(x)$ , will be $-\sin(x)$ . Multiplying these values together results in $-3\sin(3x)$ .

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Example Question #101 : How To Find Differential Functions

Find the differential of the following equation.

y=x^3+2x

Possible Answers:

dy=3x+2 dx

dy=(3x^2+2)dx

y=dx(6)

dy=(3x^2)dx

Correct answer:

dy=(3x^2+2)dx

Explanation:

To find the differential, take the derivative of each term as follows.

The derivative of anything in the form of x^n is $n{x^{n-1}}$ so applying that rule to all of the terms yields:

dy=(3x^2+2)dx

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Example Question #103 : Other Differential Functions

Find the differential of the following equation.

y=sinx+x^2

Possible Answers:

dy(sinx +2x)dx

dy=(cosx +2x)dx

dy=(cosxx^2)dx

dy=(cosx)dx

Correct answer:

dy=(cosx +2x)dx

Explanation:

The differential of is .

To find the differential of the right side of the equation, take the derivative of each term as follows.

The derivative of sin(x) is cos(x) and anything in the form of x^n is $n{x^{n-1}}$ , so applying that rule to all of the terms yields:

dy=(cosx +2x)dx

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Calculus 1 : Differential Functions

Study concepts, example questions & explanations for Calculus 1

All Calculus 1 Resources

Example Questions

Example Question #281 : Differential Functions

Example Question #282 : Differential Functions

Example Question #282 : Differential Functions

Example Question #96 : Other Differential Functions

Example Question #96 : How To Find Differential Functions

Example Question #286 : Differential Functions

Example Question #96 : How To Find Differential Functions

Example Question #101 : How To Find Differential Functions

Example Question #101 : How To Find Differential Functions

Example Question #103 : Other Differential Functions

All Calculus 1 Resources

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