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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.

# Rules of Differentiation warmup

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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