Differentiation Rules

Rules of Differentiation warmup


If f(x)=3x2,f(x) = 3x^{2}, what is the value of f(5)?f'(5)?

If f(x)=x12,f(x) = x^{\frac{1}{2}}, what is the value of f(64)?f'(64)?

Suppose f(x)=x2g(x),f(x) = x^2\cdot g(x), where g(5)=10g(5) = 10 and g(5)=4.g'(5) =4.

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

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

If f(x)=(2x)(x5),f(x) = (2x)(x^5), find f(x).f'(x).

Marla's Solution:

Apply the Product Rule: f(x)=2(x5)+2x(5x4).f'(x) = 2(x^5) + 2x(5x^4).

Paula's Solution:

(2x)(x5)=2x6, apply the Power Rule: f(x)=12x5.(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,f(x) = (10x + 1)^{50}, what is the value of f(0)?f'(0)?


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