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Differentiation Rules

Differentiation Rules Warmup

If $f(x) = 3x^2$ , which of the following is equal to $f'(5)$ ?

30 20 10 75

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by Brilliant Staff

If $f(x) = e^{x^2}$ , which of the following is equal to $f'(1)$ ?

Hint. If $f(x) = e^{x},$ then $f'(x) = e^{x}.$

e^2

e+2

2e

e

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by Brilliant Staff

If $q(x) = \frac{x^2}{e^x}$ , which of the following is equal to $q'(x)$ ?

Hint: If $g(x) = e^x$ , then $g'(x) = e^x.$

\frac{e^x(2x) + e^x(x^2)}{e^{2x}}

\frac{e^x(2x) - e^x(x^2)}{e^{2x}}

\frac{2x}{e^x}

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If $h(x) = \color{#3D99F6}{(x^3 + x^2)}\color{#D61F06}{(5x + 1)}$ , which of the following is equal to $h'(x)$ ?

$A. h'(x) = \color{#3D99F6}{(3x^2 + 2x)}\cdot \color{#D61F06}{(5)}.$

$B. h'(x) = \color{#3D99F6}{(x^3 + x^2)}\cdot \color{#D61F06}{(5x + 1)} + \color{#3D99F6}{(3x^2+2x)}\cdot \color{#D61F06}{(5)}.$

$C. h'(x) = \color{#3D99F6}{(3x^2 + 2x)}\cdot \color{#D61F06}{(5x + 1)} + \color{#3D99F6}{(x^3+x^2)}\cdot \color{#D61F06}{(5)}.$

A. B. C.

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$(f \square g)' = f' \square g'$

Which operators could replace the square to make a statement that is true for all differentiable functions $f$ and $g$ ?

\cdot \text{ only}

+ \text{ only}

+, -, \cdot , \text{ and }\div

+ \text{ and - only}

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by Brilliant Staff