d/dx[f(g(x))]
f'(g(x)) g'(x)
The chain rule states that if we have a composite function f(g(x)), then the derivative with respect to x is given by:
d/dx[f(g(x))] = f'(g(x)) * g'(x)
Here, f'(g(x)) is the derivative of f evaluated at g(x) and g'(x) is the derivative of g with respect to x.
To illustrate this rule, let’s consider an example:
Suppose we have f(x) = x^2 and g(x) = 3x – 1. Then, f(g(x)) = (3x – 1)^2.
To find the derivative of f(g(x)) with respect to x, we first find the derivative of f'(x) = 2x and evaluate it at g(x):
f'(g(x)) = 2g(x) = 2(3x – 1) = 6x – 2.
Next, we find the derivative of g(x):
g'(x) = 3.
Finally, we apply the chain rule formula:
d/dx[f(g(x))] = f'(g(x)) * g'(x) = (6x – 2) * 3 = 18x – 6.
Therefore, the derivative of f(g(x)) with respect to x is 18x – 6.
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