d/dx[f(g(x))]=f'(g(x))g'(x)
the chain rule for derivatives
In calculus, the chain rule is used to differentiate a function that is composed of two or more functions. If we have an expression of the form f(g(x)), where f and g are functions of x, and we want to find the derivative of this expression with respect to x, then we can use the chain rule.
The chain rule states that the derivative of f(g(x)) with respect to x is equal to 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. In mathematical notation, this can be written as:
d/dx[f(g(x))] = f'(g(x))g'(x)
Here, f'(g(x)) represents the derivative of the outer function f evaluated at the inner function g(x), and g'(x) represents the derivative of the inner function g(x) with respect to x.
To use the chain rule, we first differentiate the outer function f'(g(x)) with respect to its argument, namely g(x). Then, we multiply this result by the derivative of the inner function g'(x) with respect to x.
It is important to note that the chain rule can be applied to any number of composed functions. For example, if we have an expression of the form f(g(h(x))), then we can apply the chain rule twice to obtain the derivative of this expression with respect to x.
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