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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.

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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