Proof for exponential derivatives

In summary, the conversation discusses the relationship between exponential and logarithmic functions, specifically with regards to taking derivatives. It is shown that the derivative of b^x can be written as k*f'(kx), where k is a constant and f(x) is the base function. The conversation also briefly mentions the relationship between 2 and e, and how it relates to taking derivatives.
  • #1
nobahar
497
2
[tex]f(x) = 2^x \left \left[/tex]
[tex]f(kx) = 2^(kx) \left \left[/tex]
[tex]b = 2^k \left \left[/tex]
[tex]b^x = 2^(kx) \left \left[/tex]
[tex]b^x = f(kx)[/tex]
[tex]\frac{d}{dx}(b^x) = \frac{d}{dx}(f(kx)) = \frac{d}{dx}(2^(kx)) (1)[/tex]
[tex]\frac{d}{dx}(f(kx)) = k.f'(kx) (2)[/tex]
I can't see how step (1) gets to step (2).
Because I thought:
[tex]\frac{d}{dx}(f(kx)) = k.\frac{d}{dx}(f(x))[/tex]
 
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  • #2
I think it's:
[tex]\frac{d}{d(kx)}(f(kx))*\frac{d}{dx}(kx) = k*f'(kx)[/tex]
 
  • #3
d/dx(f(u)) = d/du(f(u))*du/dx
 
  • #4
nobahar said:
[tex]f(x) = 2^x \left \left[/tex]
[tex]f(kx) = 2^(kx) \left \left[/tex]
[tex]b = 2^k \left \left[/tex]
[tex]b^x = 2^(kx) \left \left[/tex]
[tex]b^x = f(kx)[/tex]
[tex]\frac{d}{dx}(b^x) = \frac{d}{dx}(f(kx)) = \frac{d}{dx}(2^(kx)) (1)[/tex]
[tex]\frac{d}{dx}(f(kx)) = k.f'(kx) (2)[/tex]
I can't see how step (1) gets to step (2).
Because I thought:
[tex]\frac{d}{dx}(f(kx)) = k.\frac{d}{dx}(f(x))[/tex]

But isn't [tex] 2=e^{\ln 2} [/tex] ? Or am i missing something ?
 
  • #5
Yes, 2 = eln 2

d/dx(f(kx)) = d/dx[(eln 2)kx] = ekxln 2 * k*ln2 = 2kx *k*ln2. The OP's formatting and organization made it a bit difficult to follow.
 
Last edited:

Related to Proof for exponential derivatives

1. What is the proof for exponential derivatives?

The proof for exponential derivatives is a mathematical process that shows how to find the derivatives of exponential functions, such as y = ex. It involves using the limit definition of derivatives and the properties of exponential functions.

2. Why is it important to understand the proof for exponential derivatives?

Understanding the proof for exponential derivatives is important because it allows us to more easily find the derivatives of more complex exponential functions. It also helps us understand the behavior and properties of exponential functions, which are commonly used in mathematical and scientific applications.

3. What are the key steps in the proof for exponential derivatives?

The key steps in the proof for exponential derivatives include using the limit definition of derivatives to find the derivative of y = ex, using the properties of logarithms to simplify the result, and then generalizing the result to any exponential function of the form y = ax, where a is a constant.

4. Can the proof for exponential derivatives be applied to other types of functions?

Yes, the proof for exponential derivatives can be applied to other types of functions, as long as they have a similar form to exponential functions. This includes functions like y = ax, where a is a constant, and y = ekx, where k is a constant.

5. Are there any common mistakes to watch out for when using the proof for exponential derivatives?

One common mistake when using the proof for exponential derivatives is forgetting to apply the chain rule when taking the derivative of y = ax. Another mistake is forgetting to use the properties of logarithms to simplify the result. It's important to carefully follow each step of the proof to avoid these and other mistakes.

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