Question

find f(x).
f(x)=5x^8 - 14x^5 + 10\sqrt{x}-\frac{5}{x^4}+12

Explanation:

Step1: Recall power - rule for differentiation

The power - rule states that if $y = ax^n$, then $y'=nax^{n - 1}$. Rewrite $\sqrt{x}=x^{\frac{1}{2}}$ and $\frac{1}{x^{4}}=x^{-4}$.

Step2: Differentiate each term

For the term $5x^{8}$, using the power - rule, its derivative is $8\times5x^{8 - 1}=40x^{7}$.
For the term $-14x^{5}$, its derivative is $5\times(-14)x^{5 - 1}=-70x^{4}$.
For the term $10\sqrt{x}=10x^{\frac{1}{2}}$, its derivative is $\frac{1}{2}\times10x^{\frac{1}{2}-1}=5x^{-\frac{1}{2}}$.
For the term $-\frac{5}{x^{4}}=-5x^{-4}$, its derivative is $(-4)\times(-5)x^{-4 - 1}=20x^{-5}$.
The derivative of the constant term $12$ is $0$.

Step3: Combine the derivatives of all terms

$f'(x)=40x^{7}-70x^{4}+5x^{-\frac{1}{2}} + 20x^{-5}$.

Answer:

$f'(x)=40x^{7}-70x^{4}+5x^{-\frac{1}{2}}+20x^{-5}$