## derivative of function

1. The problem statement, all variables and given/known data

$$y'=((3x^2+2x+5)^{8x^3+2x^2 +4})'=?$$

2. Relevant equations

3. The attempt at a solution

$$((3x^2+2x+5)^{8x^3+2x^2 +4})'=(8x^3+2x^2+4)(3x^2+2x+5)^{8x^3+2x^2 +4-1}(24x^2+4x)(6x+2)$$
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 The power rule only holds when the exponent is a constant (not a function of x).
 The function $$f(x)=g(x)^{h(x)}$$ can be written $$f(x)=e^{\ln g(x)^{h(x)}}=e^{h(x)\,\ln g(x)}$$ Now you can take the derivative, i.e. $$f'(x)=e^{h(x)\,\ln g(x)}\left(h(x)\,\ln g(x)\right)'\Rightarrow f'(x)=f(x)\left(h(x)\,\ln g(x)\right)'$$

## derivative of function

$$((3x^2+2x+5)^{8x^3+2x^2 +4})'=(3x^2+2x+5)^{8x^3+2x^2 +4}((24x^2+4x)\ln(3x^2+2x+5)+(8x^3+2x^2 +4)\frac{6x+2}{3x^2+2x+5})$$
 You missed a parethensis after $$(3x^2+2x+5)^{8x^3+2x^2 +4}$$, but you are correct

Recognitions:
Gold Member
 Quote by Rainbow Child The function $$f(x)=g(x)^{h(x)}$$ can be written $$f(x)=e^{\ln g(x)^{h(x)}}=e^{h(x)\,\ln g(x)}$$ Now you can take the derivative, i.e. $$f'(x)=e^{h(x)\,\ln g(x)}\left(h(x)\,\ln g(x)\right)'\Rightarrow f'(x)=f(x)\left(h(x)\,\ln g(x)\right)'$$