Question

hw10 differentiation rules 1 (targets c1, c5; §3.3)
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find the derivative of (f(x)=3sqrt{x}-\frac{2}{x^{7}}).
(f(x)=\frac{3}{2sqrt{x}}+\frac{4}{x^{3}}) ×
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Explanation:

Step1: Rewrite the function

Rewrite $f(x)=3\sqrt{x}-\frac{2}{x^{7}}$ as $f(x)=3x^{\frac{1}{2}} - 2x^{-7}$.

Step2: Apply power - rule for differentiation

The power - rule states that if $y = ax^{n}$, then $y^\prime=anx^{n - 1}$. For the first term $3x^{\frac{1}{2}}$, its derivative is $3\times\frac{1}{2}x^{\frac{1}{2}-1}=\frac{3}{2}x^{-\frac{1}{2}}=\frac{3}{2\sqrt{x}}$. For the second term $-2x^{-7}$, its derivative is $-2\times(-7)x^{-7 - 1}=14x^{-8}=\frac{14}{x^{8}}$.

Answer:

$f^\prime(x)=\frac{3}{2\sqrt{x}}+\frac{14}{x^{8}}$