The Chain Rule | Finding the Derivative of f(g(x)) with Respect to x

If g(x)=lnx and f is a differentiable function of x, which of the following is equivalent to the derivative of f(g(x)) with respect to x ?

To find the derivative of f(g(x)) with respect to x, we can use the chain rule

To find the derivative of f(g(x)) with respect to x, we can use the chain rule. The chain rule states that if we have a composite function f(g(x)), the derivative of this composite function with respect to x is the derivative of the outer function f evaluated at the inner function g(x), multiplied by the derivative of the inner function g(x) with respect to x.

Using this rule, we have:
(f(g(x)))’ = f'(g(x)) * g'(x)

In this case, g(x) = ln(x). So, we need to find the derivative of this function first.

The derivative of ln(x) with respect to x can be found using the basic differentiation rule for logarithmic functions. The derivative of ln(x) is 1/x.

Therefore, g'(x) = 1/x.

Now, let’s substitute this back into the chain rule equation:

(f(g(x)))’ = f'(g(x)) * g'(x)
= f'(ln(x)) * 1/x

So, the derivative of f(g(x)) with respect to x is f'(ln(x)) / x.

Therefore, the correct option is f'(ln(x)) / x.

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