How do you differentiate f(x) = (2 x^2 +5 x +10)^cos(x)f(x)=(2x2+5x+10)cos(x)?

1 Answer
Jun 4, 2016

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)