Question

find the derivative of $f(x)=5\sqrt{x}+\frac{3}{x^{10}}$.
type your answer without fractional or negative exponents. use sqrt(x) for $\sqrt{x}$.
$f(x)=$
question help: video 1 video 2

Explanation:

Step1: Rewrite the function

Rewrite \( f(x)=5\sqrt{x}+\frac{3}{x^{10}} \) as \( f(x) = 5x^{\frac{1}{2}}+3x^{- 10}\) using the rules \( \sqrt{x}=x^{\frac{1}{2}}\) and \( \frac{1}{x^{n}}=x^{-n}\).

Step2: Apply the power rule

The power rule is \( \frac{d}{dx}(x^{n})=nx^{n - 1}\).
For the first term \( y = 5x^{\frac{1}{2}}\), its derivative \( y^\prime=5\times\frac{1}{2}x^{\frac{1}{2}-1}=\frac{5}{2}x^{-\frac{1}{2}}\).
For the second term \( y = 3x^{-10}\), its derivative \( y^\prime=3\times(-10)x^{-10 - 1}=-30x^{-11}\).

Step3: Rewrite without negative exponents

\( \frac{5}{2}x^{-\frac{1}{2}}=\frac{5}{2\sqrt{x}}\) and \( - 30x^{-11}=-\frac{30}{x^{11}}\).

Answer:

\(f^{\prime}(x)=\frac{5}{2\sqrt{x}}-\frac{30}{x^{11}}\)