Question

find the derivatives for parts (a) through (d) below.
(a) if f(x)=x^8, find f(x).
f(x)=
(b) if y = x^7.1, find \frac{dy}{dx}.
\frac{dy}{dx}=
(c) if y = \frac{1}{x^8}, find dy/dx.
dy/dx=
(d) find d_x(\frac{1}{x^4}).
d_x(\frac{1}{x^4})=

Explanation:

Step1: Recall power - rule for derivatives

The power - rule states that if $y = x^n$, then $\frac{dy}{dx}=nx^{n - 1}$, where $n$ is a real number.

Step2: Solve part (a)

Given $f(x)=x^8$, using the power - rule $f^{\prime}(x)=8x^{8 - 1}=8x^{7}$.

Step3: Solve part (b)

Given $y = x^{7.1}$, by the power - rule $\frac{dy}{dx}=7.1x^{7.1 - 1}=7.1x^{6.1}$.

Step4: Rewrite part (c)

Rewrite $y=\frac{1}{x^{8}}$ as $y = x^{-8}$. Then, by the power - rule, $\frac{dy}{dx}=-8x^{-8 - 1}=-8x^{-9}=-\frac{8}{x^{9}}$.

Step5: Rewrite part (d)

Rewrite $y=\frac{1}{x^{4}}$ as $y = x^{-4}$. Then, by the power - rule, $D_x(x^{-4})=-4x^{-4 - 1}=-4x^{-5}=-\frac{4}{x^{5}}$.

Answer:

(a) $8x^{7}$
(b) $7.1x^{6.1}$
(c) $-\frac{8}{x^{9}}$
(d) $-\frac{4}{x^{5}}$