find the derivative of $f(x)=5\\sqrt{x}+\\frac{3}{x^{10}}$.\ntype your answer without fractional or negative…

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

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

Answer

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}})