Question

How do you differentiate `f(x) = (2 x^2 +5 x +10)^cos(x)`?

Answer 1

`f'(x)=((2x^2+5x+10)^cos(x)((4x+5)cos(x)-(2x^2+5x+10)sin(x)ln(2x^2+5x+10)))/(2x^2+5x+10)`

Explanation:

Let `y=(2x^2+5x+10)^cos(x)`.

Then

`ln(y)=ln((2x^2+5x+10)^cos(x))`

`ln(y)=cos(x)ln(2x^2+5x+10)`

Differentiating both sides, using the chain rule on the left and the product rule on the right:

`1/y(dy/dx)=-sin(x)ln(2x^2+5x+10)+cos(x)((4x+5)/(2x^2+5x+10))`

`1/y(dy/dx)=((4x+5)cos(x)-(2x^2+5x+10)sin(x)ln(2x^2+5x+10))/(2x^2+5x+10)`

Multiply both sides by `y`, or `(2x^2+5x+10)^cos(x)`,

`dy/dx=((2x^2+5x+10)^cos(x)((4x+5)cos(x)-(2x^2+5x+10)sin(x)ln(2x^2+5x+10)))/(2x^2+5x+10)`