Question

Find the derivative of $F(x) = e^{cos^{2}(3x)}$ using any applicable rules.

Answer 1

$f(x)=e^{\cos^2(3x)}\\\\\\f'(x)=e^{\cos^2(3x)}\cdot2\cos(3x)\cdot(-\sin (3x))\cdot 3\\\\f'(x)=-6e^{\cos^2(3x)}\sin(3x)\cos (3x)$

Answer 2

$f(x)=e^{\cos^2(3x)}\\\\f'(x)=\left(e^{\cos^2(3x)}\right)'=e^{\cos^2(3x)}\cdot2\cos(3x)\cdot(-\sin(3x))\cdot3\\\\=-3e^{\cos^2(3x)}2\sin(3x)\cos(3x)=-3e^{\cos^2(3x)}\sin(2\cdot3x)\\\\=-3e^{\cos^2(3x)}\sin(6x)\\\\Used:\\\\(e^x)'=e^x\\\\\left(f(g(x))\right)'=f'(g(x))\cdot g'(x)\\\\(x^n)'=nx^{n-1}\\\\\sin(2x)=2\sin x\cos x$