How To Use The Chain Rule To Find 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 ?

Correct. The chain rule provides a way to differentiate a composition of functions. The outside function is f(x)f(x) and the inside function is g(x)=lnxg(x)=ln⁡x. Then g′(x)=1xg′(x)=1x and the chain rule can be used, as follows. ddxf(lnx)=ddxf(g(x))=f′(g(x))⋅g′(x)=f′(lnx)⋅1x

We can use the chain rule to differentiate f(g(x)) with respect to x.

Let u = g(x) = ln x, then du/dx = 1/x (using the derivative of natural logarithm).

Applying the chain rule, we get:

df/dx = df/du * du/dx

df/dx = f'(u) * 1/x

df/dx = f'(ln x) / x

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

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