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 composition of functions, such as f(g(x)), then the derivative of this composition is the derivative of the outer function evaluated at the inner function, multiplied by the derivative of the inner function.
In this case, the outer function is f, and the inner function is g(x) = ln(x).
Using the chain rule, we can write:
(f(g(x)))’ = f'(g(x)) * g'(x)
To find the derivative of f(g(x)), we need to know the derivative of f with respect to g(x), denoted as f'(g(x)), and the derivative of g(x) with respect to x, denoted as g'(x).
Since f is a differentiable function of x, we have f'(x) available. However, to find f'(g(x)), we need to apply the chain rule again.
Let’s denote h(x) = f'(x). Then, the derivative of f(g(x)) with respect to x becomes:
(f(g(x)))’ = h(g(x)) * g'(x)
So, the expression equivalent to the derivative of f(g(x)) with respect to x is h(g(x)) * g'(x).
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