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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 \(f(x) = 3x^{2},\) what is the value of \(f'(5)?\)

If \[f(x) = x^{\frac{1}{2}},\] what is the value of \(f'(64)?\)

Suppose \[f(x) = x^2\cdot g(x),\] where \(g(5) = 10\) and \(g'(5) =4.\)

What is the value of \(f'(5)?\)

Marla and Paula are confronted with the following question on their calculus exam:

If \(f(x) = (2x)(x^5),\) find \(f'(x).\)

Marla's Solution:

Apply the Product Rule: \[f'(x) = 2(x^5) + 2x(5x^4).\]

Paula's Solution:

\[(2x)(x^5) = 2x^6 \mbox{, apply the Power Rule: } f'(x) = 12x^5.\]

Who got it right?

**Note.** The grader does not require the answer to be fully simplified.

If \(f(x) = (10x + 1)^{50},\) what is the value of \(f'(0)?\)

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