Q. 43 - 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.  

$f\left(x\right)={\left({\left(2x+1\right)}^{-5}-1\right)}^{-9}$ 

Step-by-Step Solution

Answer

The required answer is $-9{\left({\left(2x+1\right)}^{-5}-1\right)}^{-10}(-10{\left(2x+1\right)}^{-6})$

Step 1. Given Information

The given function is $f\left(x\right)={\left({\left(2x+1\right)}^{-5}-1\right)}^{-9}$

Step 2. Calculation

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

$$\begin{gathered} f\text{'}\left(x\right)=-9{\left({\left(2x+1\right)}^{-5}-1\right)}^{-9-1}\frac{d}{dx}\left({\left(2x+1\right)}^{-5}-1\right) \\ =-9{\left({\left(2x+1\right)}^{-5}-1\right)}^{-10}\left(\frac{d}{dx}{\left(2x+1\right)}^{-5}-\frac{d}{dx}1\right) \\ =-9{\left({\left(2x+1\right)}^{-5}-1\right)}^{-10}\left(-5{\left(2x+1\right)}^{-5-1}\frac{d}{dx}(2x+1)\right) \\ =-9{\left({\left(2x+1\right)}^{-5}-1\right)}^{-10}\left(-5{\left(2x+1\right)}^{-6}\left(2\right)\right) \\ =-9{\left({\left(2x+1\right)}^{-5}-1\right)}^{-10}\left(-10{\left(2x+1\right)}^{-6}\right) \end{gathered}$$