Question

Differentiate y= (x²-x)^9(√5x²+4x)

Answer 1

The given expression is

= (x²-x)^9(√5x²+4x​)

We would differentiate it by applying the product rule. It is expressed as

(fg)' = f'g + fg'

Let f = (x²-x)^9

Let g = (√5x²+4x​)

f' = 9(2x - 1)(x²-x)^8

g = (√5x²+4x​) = (5x²+4x)^1/2

g' = 1/2(10x + 4)(√5x²+4x​)^-1/2

g' = (10x + 4)/2(√5x²+4x)

g' = (5x + 2)/(√5x²+4x)

By applying the rule, it becomes

9(2x - 1)(x²-x)^8 * (√5x²+4x​) + (x²-x)^9 * (5x + 2)/(√5x²+4x)

By simplifying,

`y^(\prime)\text{ = }\frac{5x(x^2-x)^9+\text{ 2\lparen x}^2-x)^9+\text{ 90x}^3(x^2-x)^8+27x^2(x^2-x)^8-36x(x^2-x)^8}{\sqrt{5x^2\text{ + 4x}}}`