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

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Answer 1 · 2016-06-04T15:21:28

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

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 $(2x^2+5x+10)^cos(x)$ ,

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

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