Chain Rule
The chain rule is a fundamental rule in calculus for differentiating composite functions. It allows us to break complex derivatives into simpler parts.
Formula
If y = f(g(x)), then
dy/dx = f'(g(x)) · g'(x)
where f' is the derivative of f with respect to its argument, and g' is the derivative of g with respect to x.
How to Apply
Step 1: Identify inner and outer functions
Determine the structure of the composite function: outer function f(u) and inner function u = g(x).
Step 2: Differentiate each part
Compute f'(u) with respect to u and g'(x) with respect to x.
Step 3: Multiply to get the result
Multiply the two derivatives: f'(g(x)) · g'(x).
Practice Problems
Difficulty: Beginner
d/dx [ (x^2 + 1)^3 ]
Difficulty: Intermediate
d/dx [ sin(2x^2) ]
Difficulty: Intermediate
d/dx [ e^(3x^2 + 2x) ]
Difficulty: Advanced
d/dx [ ln(cos(x)) ]
Difficulty: Advanced
d/dx [ (x^3 + 2x)^(1/2) ]
Related Calculators
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