Question

hw10 differentiation rules 1 (targets c1, c5; §3.3)
score: 0/9 answered: 0/9
question 1
(1) if (f(x)=x^{8}), then (f^{prime}(x)=)
(2) if (g(x)= - 4x^{5}), then (g^{prime}(x)=)
(3) if (h(x)=\frac{1}{x^{3}}), then (h^{prime}(x)=)
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Explanation:

Step1: Apply power - rule for $f(x)$

The power - rule for differentiation is $\frac{d}{dx}(x^n)=nx^{n - 1}$. For $f(x)=x^8$, using the power - rule, we have $f'(x)=8x^{8 - 1}=8x^{7}$.

Step2: Apply power - rule for $g(x)$

For $g(x)=-4x^{5}$, by the power - rule $\frac{d}{dx}(ax^n)=anx^{n - 1}$ (where $a=-4$ and $n = 5$), we get $g'(x)=-4\times5x^{5 - 1}=-20x^{4}$.

Step3: Rewrite $h(x)$ and apply power - rule

Rewrite $h(x)=\frac{1}{x^{3}}=x^{-3}$. Then, using the power - rule $\frac{d}{dx}(x^n)=nx^{n - 1}$, we have $h'(x)=-3x^{-3 - 1}=-3x^{-4}=-\frac{3}{x^{4}}$.

Answer:

(1) $8x^{7}$
(2) $-20x^{4}$
(3) $-\frac{3}{x^{4}}$