Explanation
The chain rule is used to differentiate a composite function. A composite function consists of an outer function and an inner function. The chain rule states that you first differentiate the outer function and then multiply by the derivative of the inner function. The chain rule formula is:
Procedure
To differentiate a composite function, follow these steps:
1. Determine the outer function u and the inner function v.
2. Differentiate the outer function u with respect to v.
3. Differentiate the inner function v with respect to x.
4. Multiply both derivatives together to get the derivative of the composite function.
Examples
Differentiate the function f(x) = (3x + 2)^2 .
Outer function: u(v) = v^2, inner function: v(x) = 3x + 2.
u'(v) = 2v, v'(x) = 3.
Differentiate the function f(x) = \sin(4x^2) .
Outer function: u(v) = \sin(v), inner function: v(x) = 4x^2.
u'(v) = \cos(v), v'(x) = 8x.
Note
- ★The chain rule is used to differentiate composite functions.
- ★Derivative of the outer function multiplied by the derivative of the inner function gives the overall derivative.
Exercises
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