Understanding d/dx Notation and Calculating the Derivative of f(g(x))

d/dx f(g(x))

To answer your question, let’s start with a brief explanation of the notation “d/dx”

To answer your question, let’s start with a brief explanation of the notation “d/dx”. The symbol “d/dx” represents the derivative operator, which is used to find the rate at which a function changes with respect to its independent variable, in this case, “x”.

Now, let’s dive into “f(g(x))”. In this notation, we have a composition of two functions, namely “f(x)” and “g(x)”. The function “g(x)” is being used as an input into the function “f(x)”.

To calculate the derivative of “f(g(x))” with respect to “x”, we need to apply the chain rule. The chain rule enables us to find the derivative of a composite function by breaking it down into simpler parts.

The chain rule states that if we have a function “h(x)” which is a composition of two functions, say “f(u)” and “u(g(x))”, then the derivative of “h(x)” with respect to “x” can be calculated as follows:

dh/dx = (dh/du) * (du/dx)

Applying the chain rule to “f(g(x))”, we have:

d/dx f(g(x)) = (df/du) * (du/dx)

Here, “(df/du)” represents the derivative of the outer function “f(u)” with respect to its independent variable “u”, and “(du/dx)” represents the derivative of the inner function “u(g(x))” with respect to its independent variable “x”.

In summary, to find the derivative of “f(g(x))” with respect to “x”, you need to take the derivative of the outer function “f(u)” with respect to its variable, multiply it by the derivative of the inner function “u(g(x))” with respect to “x”.

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