Q. 22 - Calculus

Question

Find the derivatives of the functions in Exercises 21–46. Keep in mind that it may be convenient to do some preliminary algebra before differentiating.

$f\left(x\right)={\left(\sqrt{x}+4\right)}^5$

Step-by-Step Solution

Answer

The required answer is $\frac{5{\left(\sqrt{x}+4\right)}^4}{2\sqrt{x}}$

Step 1. Given Information

The given function is $f\left(x\right)={\left(\sqrt{x}+4\right)}^5$

Step 2. Calculation

Differentiate both the sides with respect to x, we get,

$$\begin{gathered} f\text{'}\left(x\right)=\frac{d}{dx}{\left(\sqrt{x}+4\right)}^5 \\ =5{\left(\sqrt{x}+4\right)}^{5-1}\frac{d}{dx}\left(\sqrt{x}+4\right) \\ =5{(\sqrt{x}+4)}^4\left(\frac{d}{dx}\sqrt{x}+\frac{d}{dx}4\right) \\ =5{(\sqrt{x}+4)}^4\left(\frac{1}{2}{x}^{\frac{1}{2}-1}\right) \\ =5{(\sqrt{x}+4)}^4\left(\frac{1}{2}{x}^{-\frac{1}{2}}\right) \\ =\frac{5{(\sqrt{x}+4)}^4}{2\sqrt{x}} \end{gathered}$$