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# Differentiation Rules

These are the rules that explain how to take derivatives of any functions: from polynomials to trigonometric functions to logarithms.

# Differentiation Rules Warmup

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

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

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

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

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

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

$A. h'(x) = \color{blue}{(3x^2 + 2x)}\cdot \color{red}{(5)}.$

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

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

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

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

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