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Calculus 1 : How to find differential functions

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Example Questions

Example Question #91 : How To Find Differential Functions

Differentiate the function

$f(x) = \sin(\cos(2x))$

Possible Answers:

$f(x) = -2\cos(\cos(2x))\sin(2x)$

None of these answers are correct.

$f(x) = 2\cos(\cos(2x))\sin(2x)$

$f(x) = \cos(\cos(2x))\sin(2x)$

Correct answer:

$f(x) = -2\cos(\cos(2x))\sin(2x)$

Explanation:

To differentiate the function properly, we must use the Chain Rule which is,

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

Therefore the derivative of the function is,

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

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Example Question #92 : How To Find Differential Functions

Differentiate

$f(x) = \sin^2(x^2)$

Possible Answers:

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

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

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

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

Correct answer:

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

Explanation:

To differentiate this equation we use the Chain Rule.

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

Using this throughout the equation gives us,

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

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Example Question #93 : How To Find Differential Functions

Find the derivative of

$f(x) = 4\sin(x+3x^2)$

Possible Answers:

$\small f'(x) = 4\cos(x+3x^2)(1+6x)$

$\small f'(x) = -4\cos(x+3x^2)(1+6x)$

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

None of these answers are correct.

Correct answer:

$\small f'(x) = 4\cos(x+3x^2)(1+6x)$

Explanation:

To find the derivative of the function we must use the Chain Rule, which is

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

Applying this to the function we get,

$f'(x) = 4\cos(x+3x^2)(1+6x)$

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Example Question #94 : How To Find Differential Functions

Find the derivative of

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

Possible Answers:

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

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

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

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

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 #95 : How To Find Differential Functions

Differentiate the following function

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

Possible Answers:

None of these answers are correct.

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

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

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

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 #284 : Differential Functions

Find the derivative of

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

Possible Answers:

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

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

None of these answers are correct.

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

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 #97 : 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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Calculus 1 : How to find differential functions

Study concepts, example questions & explanations for Calculus 1

All Calculus 1 Resources

Example Questions

Example Question #91 : How To Find Differential Functions

Example Question #92 : How To Find Differential Functions

Example Question #93 : How To Find Differential Functions

Example Question #94 : How To Find Differential Functions

Example Question #282 : Differential Functions

Example Question #95 : How To Find Differential Functions

Example Question #284 : Differential Functions

Example Question #96 : How To Find Differential Functions

Example Question #286 : Differential Functions

Example Question #97 : How To Find Differential Functions

All Calculus 1 Resources

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