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 \(f'(0)?\)