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.

$f\left(x\right)={\left(1-4x\right)}^2{(3x^2+1)}^9$

Short answer

The required answer is$(1-4x){(3x^2+1)}^8(-8(3x^2+1)+54x(1-4x\left)\right)$

Step-by-step solution

Step 1. Given Information   

The given function is $f\left(x\right)={\left(1-4x\right)}^2{(3x^2+1)}^9$

Step 2. Calculation    

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

$$\begin{gathered} f\text{'}\left(x\right)=\frac{d}{dx}{\left(1-4x\right)}^2{(3x^2+1)}^9 \\ =\left(\frac{d}{dx}{\left(1-4x\right)}^2\right){(3x^2+1)}^9+{\left(1-4x\right)}^2\left(\frac{d}{dx}{(3x^2+1)}^9\right) \\ =\left(2(1-4x)(-4)\right){(3x^2+1)}^9+{\left(1-4x\right)}^2\left(9{(3x^2+1)}^{8}3\left(2x\right)\right) \\ =-8(1-4x){(3x^2+1)}^9+54x{(3x^2+1)}^8{(1-4x)}^2 \\ =(1-4x){(3x^2+1)}^8\left(-8(3x^2+1)+54x(1-4x)\right) \end{gathered}$$