find the derivative of f(x)= 2^cot^2(3x^3+5)base e^-5x^2. Hint: cot^2(x)=(cot(x))^2 (Do not simplify!) (Step by step)

Question

find the derivative of f(x)= 2^cot^2(3x^3+5)base e^-5x^2.   Hint: cot^2(x)=(cot(x))^2  (Do not simplify!) (Step by step)

freckles · Answer

what is that base thing going on there?

anonymous · Answer

$$f(x)=2^{\cot ^{2}(3x ^{3}+5)_e ^{-5x ^{2}}}$$

freckles · Answer

$$\ f(x)=2^{\cot^2(3x^3+5)e^{-5x^2}} \ \text{ Let } y=2^{\cot^2(3x^3+5)e^{-5x^2}} \ \ln(y)=\cot^2(3x^3+5)e^{-5x^2} \ln(2)$$
Now differentiate both sides

anonymous · Answer

$$y'(x)=xy \log({2})(\sin(6x ^{3}+10)-18x ^{3})\cot (3x ^{3}+5)\csc ^{2}(3x ^{3}+5) \div e ^{5}$$            
@freckles

freckles · Answer

$$\frac{y'}{y}=\ln(2) \cdot ([\cot^2(3x^3+5)]'e^{-5x^2}+\cot^2(3x^3+5) [e^{-5x^2}]' )$$

freckles · Answer

$$[\cot^2(3x^3+5)]'=(3x^3+5)' \cdot 2 \cot^{2-1}(3x^3+5) \cdot (-\csc^2(3x^3+5)) \ [e^{-5x^2}]'=(-5x^2)'e^{-5x^2}$$