Calculus
# Differentiation Rules

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 $f'(0)?$