What is the derivative of a function raised to another function?

In summary, the derivative of y'=((3x^2+2x+5)^{8x^3+2x^2 +4}) is (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}).
  • #1
fermio
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0

Homework Statement



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

Homework Equations





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]
 
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  • #2
The power rule only holds when the exponent is a constant (not a function of x).
 
Last edited:
  • #3
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]
 
  • #4
[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]
 
  • #5
You missed a parethensis after [tex](3x^2+2x+5)^{8x^3+2x^2 +4}[/tex], but you are correct :smile:
 
  • #6
Rainbow Child said:
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).
 

What is the derivative of (3x^2+2x+5)^n?

The derivative of (3x^2+2x+5)^n is n(3x^2+2x+5)^(n-1)*(6x+2), which can be simplified to n(6x^3+9x^2+6x+2).

How do you take the derivative of (3x^2+2x+5)^n?

To take the derivative of (3x^2+2x+5)^n, you can use the power rule and the chain rule. First, bring the exponent down and subtract one from it. Then, multiply the derivative of the inside function by the original function to the power of n-1.

What is the significance of the exponent in the derivative of (3x^2+2x+5)^n?

The exponent in the derivative of (3x^2+2x+5)^n represents the number of times the original function is being multiplied by itself. This is known as the power rule and is used to find the derivatives of exponential functions.

Can the exponent in the derivative of (3x^2+2x+5)^n be negative?

Yes, the exponent in the derivative of (3x^2+2x+5)^n can be negative. This would result in a negative power rule, where the exponent is brought down and subtracted by one, and the negative sign is kept in the derivative.

How does the exponent affect the shape of the graph of (3x^2+2x+5)^n?

The exponent in the derivative of (3x^2+2x+5)^n affects the steepness of the graph. The larger the exponent, the steeper the graph will be. On the other hand, a smaller exponent will result in a flatter graph. Additionally, a negative exponent will cause the graph to be reflected over the x-axis.

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