Register to reply

Derivative of function

by fermio
Tags: derivative, function
Share this thread:
fermio
#1
Jan23-08, 01:46 AM
P: 38
1. The problem statement, all variables and given/known data

[tex]y'=((3x^2+2x+5)^{8x^3+2x^2 +4})'=?[/tex]

2. Relevant equations



3. The attempt at a solution

[tex]((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)[/tex]
Phys.Org News Partner Science news on Phys.org
Scientists develop 'electronic nose' for rapid detection of C. diff infection
Why plants in the office make us more productive
Tesla Motors dealing as states play factory poker
ehrenfest
#2
Jan23-08, 01:55 AM
P: 1,996
The power rule only holds when the exponent is a constant (not a function of x).
Rainbow Child
#3
Jan23-08, 02:02 AM
P: 365
The function [tex]f(x)=g(x)^{h(x)}[/tex] can be written

[tex]f(x)=e^{\ln g(x)^{h(x)}}=e^{h(x)\,\ln g(x)}[/tex]

Now you can take the derivative, i.e.

[tex]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)'[/tex]

fermio
#4
Jan23-08, 02:22 AM
P: 38
Derivative of function

[tex]((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})[/tex]
Rainbow Child
#5
Jan23-08, 02:25 AM
P: 365
You missed a parethensis after [tex](3x^2+2x+5)^{8x^3+2x^2 +4}[/tex], but you are correct
HallsofIvy
#6
Jan23-08, 06:25 AM
Math
Emeritus
Sci Advisor
Thanks
PF Gold
P: 39,564
Quote Quote by Rainbow Child View Post
The function [tex]f(x)=g(x)^{h(x)}[/tex] can be written

[tex]f(x)=e^{\ln g(x)^{h(x)}}=e^{h(x)\,\ln g(x)}[/tex]

Now you can take the derivative, i.e.

[tex]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)'[/tex]
Or, much the same thing, write ln(f(x))= h(x)ln(g(x)) and use the product and chain rules: (1/f)f '= h'(x) ln(g(x))+ (h(x)/g(x)) g'(x) so f '= [h'(x) ln(g(x)+(h(x)/g(x))g'(x)]f(x).


Register to reply

Related Discussions
Need Help With Derivative Function Calculus & Beyond Homework 6
Lie derivative of a function Differential Geometry 5
Derivative of a function with ln Calculus & Beyond Homework 7
Derivative w.r.t. a function Calculus 9
Derivative of the function Calculus 4