d/dx[f(g(x))]=f'(g(x))g'(x)
The given equation is known as the chain rule in calculus
The given equation is known as the chain rule in calculus. It is used to find the derivative of a composition of functions. Let’s break down the equation step by step:
1. Let’s consider two functions, f(x) and g(x), where f(x) is a function of g(x). So, we can rewrite f(g(x)) as F(x) = f(g(x)).
2. Now, we want to find the derivative of F(x) with respect to x, which is denoted as dF/dx.
3. The right-hand side of the equation, f'(g(x))g'(x), represents the derivative of f(g(x)) with respect to x. Here, f'(g(x)) represents the derivative of f(x) with respect to g(x), evaluated at g(x), and g'(x) represents the derivative of g(x) with respect to x.
4. When applying the chain rule, we follow a two-step process:
a. First, we find the derivative of f(g(x)) with respect to g(x), which is denoted as df/dg. This step is equivalent to taking the derivative of the outer function f(x) while treating g(x) as the independent variable. So, df/dg represents the derivative of f(x) with respect to x, and we simply replace x with g(x). This derivative is evaluated at g(x).
b. Second, we multiply df/dg by g'(x), which represents the derivative of g(x) with respect to x. This step is equivalent to taking the derivative of the inner function g(x) with respect to x.
5. Combining the two steps described above, we get dF/dx = df/dg * g'(x) = f'(g(x))g'(x), which is the chain rule formula.
In summary, the chain rule allows us to find the derivative of a composition of functions by taking the derivative of the outer function with respect to the inner function and then multiplying it by the derivative of the inner function with respect to the variable. This rule is particularly useful in calculus when dealing with complex functions that are composed of multiple simpler functions.
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