DerivativeCalculus.com

Chain Rule Mastery Worksheet

Master composite function differentiation with 10 comprehensive problems and step-by-step solutions

📚 Understanding the Chain Rule
d/dx [f(g(x))] = f'(g(x)) · g'(x)
Problem 1 Beginner
y = (3x² + 5)⁴
1 Identify inner and outer functions:
Outer: f(u) = u⁴, Inner: g(x) = 3x² + 5
2 Differentiate outer function (power rule):
f'(u) = 4u³
3 Differentiate inner function:
g'(x) = 6x
4 Apply chain rule: f'(g(x)) · g'(x)
y' = 4(3x² + 5)³ · 6x
5 Simplify:
y' = 24x(3x² + 5)³
y' = 24x(3x² + 5)³
Problem 2 Beginner
y = sin(4x³)
1 Identify inner and outer functions:
Outer: f(u) = sin(u), Inner: g(x) = 4x³
2 Differentiate outer function:
f'(u) = cos(u)
3 Differentiate inner function:
g'(x) = 12x²
4 Apply chain rule:
y' = cos(4x³) · 12x²
5 Simplify:
y' = 12x²cos(4x³)
y' = 12x²cos(4x³)
Problem 3 Intermediate
y = e^(x² + 1)
1 Identify inner and outer functions:
Outer: f(u) = e^u, Inner: g(x) = x² + 1
2 Differentiate outer function:
f'(u) = e^u
3 Differentiate inner function:
g'(x) = 2x
4 Apply chain rule:
y' = e^(x² + 1) · 2x
5 Simplify:
y' = 2xe^(x² + 1)
y' = 2xe^(x² + 1)
Problem 4 Intermediate
y = ln(3x - 2)
1 Identify inner and outer functions:
Outer: f(u) = ln(u), Inner: g(x) = 3x - 2
2 Differentiate outer function:
f'(u) = 1/u
3 Differentiate inner function:
g'(x) = 3
4 Apply chain rule:
y' = (1/(3x - 2)) · 3
5 Simplify:
y' = 3/(3x - 2)
y' = 3/(3x - 2)
Problem 5 Advanced
y = √(cos(x))
1 Rewrite square root as power:
y = (cos(x))^(1/2)
2 Identify inner and outer functions:
Outer: f(u) = u^(1/2), Inner: g(x) = cos(x)
3 Differentiate outer function (power rule):
f'(u) = (1/2)u^(-1/2) = 1/(2√u)
4 Differentiate inner function:
g'(x) = -sin(x)
5 Apply chain rule:
y' = (1/(2√(cos(x)))) · (-sin(x))
6 Simplify:
y' = -sin(x)/(2√(cos(x)))
y' = -sin(x)/(2√(cos(x)))
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