# Derivative of function

by fermio
Tags: derivative, function
 P: 38 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 $$((3x^2+2x+5)^{8x^3+2x^2 +4})'=(8x^3+2x^2+4)(3x^2+2x+5)^{8x^3+2x^2 +4-1}(24x^2+4x)(6x+2)$$
 P: 1,996 The power rule only holds when the exponent is a constant (not a function of x).
 P: 365 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. $$f'(x)=e^{h(x)\,\ln g(x)}\left(h(x)\,\ln g(x)\right)'\Rightarrow f'(x)=f(x)\left(h(x)\,\ln g(x)\right)'$$
 P: 38 Derivative of function $$((3x^2+2x+5)^{8x^3+2x^2 +4})'=(3x^2+2x+5)^{8x^3+2x^2 +4}((24x^2+4x)\ln(3x^2+2x+5)+(8x^3+2x^2 +4)\frac{6x+2}{3x^2+2x+5})$$
 P: 365 You missed a parethensis after $$(3x^2+2x+5)^{8x^3+2x^2 +4}$$, but you are correct
Math
Emeritus
 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. $$f'(x)=e^{h(x)\,\ln g(x)}\left(h(x)\,\ln g(x)\right)'\Rightarrow f'(x)=f(x)\left(h(x)\,\ln g(x)\right)'$$