The is a powerful tool for finding derivatives of composite functions. It allows us to break down complex functions into simpler parts, making differentiation easier. This rule is essential for tackling a wide range of mathematical problems in calculus.
Combining the chain rule with other differentiation techniques expands our problem-solving toolkit. From power and product rules to multiple compositions, the chain rule's versatility shines through. Its real-world applications in physics, economics, and engineering make it a crucial concept to master.
The Chain Rule
Chain rule for composite functions
Top images from around the web for Chain rule for composite functions
OpenAlgebra.com: Free Algebra Study Guide & Video Tutorials: Function Composition View original
Apply chain rule if f(x) or g(x) is composite function
Examples:
h(x)=(x2+1)3, power rule and chain rule
h(x)=sin(x)⋅ex, product rule and chain rule
Chain rule for multiple compositions
Applies to compositions of three or more functions h(x)=f(g(k(x)))
Derivative: h′(x)=f′(g(k(x)))⋅g′(k(x))⋅k′(x)
Work from outside in, applying chain rule at each step
Multiply derivatives of each function in
Example: h(x)=ln(sin(ex))
Outer function f(x)=ln(x), middle function g(x)=sin(x), inner function k(x)=ex
h′(x)=sin(ex)1⋅cos(ex)⋅ex
Mathematical basis of chain rule
Based on concept of composite function with "inner" and "outer" functions
Chain rule calculates how changes in x affect g(x) and then how changes in g(x) affect f(g(x))
Justified using limit definition of derivative
Applying limit definition to composite function shows derivative is product of outer and inner function derivatives
Real-world applications of chain rule
Useful for solving problems involving rates of change in physics, economics, engineering
Velocity and acceleration:
Position s(t), velocity v(t)=s′(t), acceleration a(t)=v′(t)
Use chain rule if s(t) is composite function
Marginal cost and revenue in economics:
Cost C(x), revenue R(x), use chain rule if C(x) or R(x) are composite functions
Optimization problems in engineering:
Objective function or constraints involve composite functions
Chain rule helps find optimal solution by calculating necessary derivatives
Variables and Implicit Differentiation
Dependent variable: The output of a function, typically y or f(x)
Independent variable: The input of a function, typically x
Implicit differentiation: A technique using the chain rule to find derivatives of implicitly defined functions
Useful when a function is not explicitly solved for the dependent variable
Key Terms to Review (3)
Chain rule: The chain rule is a formula for computing the derivative of the composition of two or more functions. It states that if $y = f(g(x))$, then the derivative $dy/dx = f'(g(x)) * g'(x)$.
Composition: Composition is the act of combining or arranging multiple elements, functions, or operations into a unified whole. It is a fundamental concept in mathematics and various fields, describing how different components interact and integrate to form a cohesive structure or process.
Function Composition: Function composition is the process of combining two or more functions to create a new function. The resulting function represents the combined effect of applying the individual functions in a specific order.