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Let k be a constant. Compute d/dx [In(kx)] in two ways. Using the Chain Rule, first decompose y = ln(kx) into an outside and inside function.

Outside function (in terms of u): y = __________

Respuesta :

facundo3141592

The differentiation using the chain rule gives:

[tex]\frac{d(g(f(x)))}{dt} = g'(f(x))*f'(x) = \frac{1}{(kx)}*k = \frac{1}{x}[/tex]

How to differentiate a logarithmic function?

Remember that:

[tex]ln(x) = \int\limits {\frac{1}{x} } \, dx[/tex]

Now, we want to derivate ln(kx) where k is a constant, the problem tells us to use the chain rule, so we define two functions:

  • g(x) = ln(x)
  • f(x) = kx

Then we have:

g(f(x))

Remember that the chain rule says that:

[tex]\frac{d(g(f(x)))}{dt} = g'(f(x))*f'(x)[/tex]

In this case:

  • g'(x) = 1/x
  • f'(x) = k

Then:

[tex]\frac{d(g(f(x)))}{dt} = g'(f(x))*f'(x) = \frac{1}{(kx)}*k = \frac{1}{x}[/tex]

If you want to learn more about the chain rule, you can read:

https://brainly.com/question/26731132

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