# Evaluate {f}'(x) at the following points: a) {f}'(3) b) {f}'(5)Let f(x)=5x^{{1/4}(x^{3}-10).

## Question:

Let {eq}f(x)=5x^{1/4}(x^{3}-10). {/eq} Evaluate {eq}{f}'(x) {/eq} at the following points:

a) {eq}{f}'(3) {/eq}

b) {eq}{f}'(5) {/eq}

## Evaluating the Derivative:

f'(x) is the derivative of f(x). Therefore, we first calculate the derivative of the given function. After this, we find its value at the two points given. These values give the rate of change of f(x) at the particular x-value.

We first need to find f'(x).

{eq}\displaystyle f'(x)=\frac{\mathrm{d} }{\mathrm{d} x}\left [5x^{1/4}(x^{3}-10) \right ]\\ \displaystyle =\frac{\mathrm{d} }{\mathrm{d} x}(5x^{13/4}-50x^{1/4})\\ \displaystyle =5*\frac{13}{4}*x^{\frac{13}{4}-1}-50*\frac{1}{4}*x^{\frac{1}{4}-1}\\ \displaystyle \therefore f'(x)=16.25x^{2.25}-12.5x^{-0.75} {/eq}

a) {eq}f'(3)=16.25*3^{2.25}-12.5*3^{-0.75}\\=186.992 {/eq}

b) {eq}f'(5)=16.25*5^{2.25}-12.5*5^{-0.75}\\=603.747 {/eq}