The chain rule 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
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Finds derivative of composite function f ( g ( x ) ) f(g(x)) f ( g ( x )) (function composition )
Identifies outer function f f f and inner function g g g
Multiplies derivative of outer function f ′ ( g ( x ) ) f'(g(x)) f ′ ( g ( x )) by derivative of inner function g ′ ( x ) g'(x) g ′ ( x )
Final derivative: h ′ ( x ) = f ′ ( g ( x ) ) ⋅ g ′ ( x ) h'(x) = f'(g(x)) \cdot g'(x) h ′ ( x ) = f ′ ( g ( x )) ⋅ g ′ ( x )
Examples:
h ( x ) = sin ( x 2 ) h(x) = \sin(x^2) h ( x ) = sin ( x 2 ) , outer function f ( x ) = sin ( x ) f(x) = \sin(x) f ( x ) = sin ( x ) , inner function g ( x ) = x 2 g(x) = x^2 g ( x ) = x 2
h ( x ) = e cos ( x ) h(x) = e^{\cos(x)} h ( x ) = e c o s ( x ) , outer function f ( x ) = e x f(x) = e^x f ( x ) = e x , inner function g ( x ) = cos ( x ) g(x) = \cos(x) g ( x ) = cos ( x )
Calculates the rate of change of nested functions
Combining chain rule with other rules
Power rule: h ( x ) = ( g ( x ) ) n h(x) = (g(x))^n h ( x ) = ( g ( x ) ) n , h ′ ( x ) = n ( g ( x ) ) n − 1 ⋅ g ′ ( x ) h'(x) = n(g(x))^{n-1} \cdot g'(x) h ′ ( x ) = n ( g ( x ) ) n − 1 ⋅ g ′ ( x )
Product rule: h ( x ) = f ( x ) ⋅ g ( x ) h(x) = f(x) \cdot g(x) h ( x ) = f ( x ) ⋅ g ( x ) , h ′ ( x ) = f ′ ( x ) ⋅ g ( x ) + f ( x ) ⋅ g ′ ( x ) h'(x) = f'(x) \cdot g(x) + f(x) \cdot g'(x) h ′ ( x ) = f ′ ( x ) ⋅ g ( x ) + f ( x ) ⋅ g ′ ( x )
Apply chain rule if f ( x ) f(x) f ( x ) or g ( x ) g(x) g ( x ) is composite function
Quotient rule: h ( x ) = f ( x ) g ( x ) h(x) = \frac{f(x)}{g(x)} h ( x ) = g ( x ) f ( x ) , h ′ ( x ) = f ′ ( x ) ⋅ g ( x ) − f ( x ) ⋅ g ′ ( x ) ( g ( x ) ) 2 h'(x) = \frac{f'(x) \cdot g(x) - f(x) \cdot g'(x)}{(g(x))^2} h ′ ( x ) = ( g ( x ) ) 2 f ′ ( x ) ⋅ g ( x ) − f ( x ) ⋅ g ′ ( x )
Apply chain rule if f ( x ) f(x) f ( x ) or g ( x ) g(x) g ( x ) is composite function
Examples:
h ( x ) = ( x 2 + 1 ) 3 h(x) = (x^2 + 1)^3 h ( x ) = ( x 2 + 1 ) 3 , power rule and chain rule
h ( x ) = sin ( x ) ⋅ e x h(x) = \sin(x) \cdot e^x h ( x ) = sin ( x ) ⋅ e x , product rule and chain rule
Chain rule for multiple compositions
Applies to compositions of three or more functions h ( x ) = f ( g ( k ( x ) ) ) h(x) = f(g(k(x))) h ( x ) = f ( g ( k ( x )))
Derivative: h ′ ( x ) = f ′ ( g ( k ( x ) ) ) ⋅ g ′ ( k ( x ) ) ⋅ k ′ ( x ) h'(x) = f'(g(k(x))) \cdot g'(k(x)) \cdot k'(x) 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 composition
Example: h ( x ) = ln ( sin ( e x ) ) h(x) = \ln(\sin(e^x)) h ( x ) = ln ( sin ( e x ))
Outer function f ( x ) = ln ( x ) f(x) = \ln(x) f ( x ) = ln ( x ) , middle function g ( x ) = sin ( x ) g(x) = \sin(x) g ( x ) = sin ( x ) , inner function k ( x ) = e x k(x) = e^x k ( x ) = e x
h ′ ( x ) = 1 sin ( e x ) ⋅ cos ( e x ) ⋅ e x h'(x) = \frac{1}{\sin(e^x)} \cdot \cos(e^x) \cdot e^x h ′ ( x ) = s i n ( e x ) 1 ⋅ cos ( e x ) ⋅ e x
Mathematical basis of chain rule
Based on concept of composite function with "inner" and "outer" functions
Chain rule calculates how changes in x x x affect g ( x ) g(x) g ( x ) and then how changes in g ( x ) g(x) g ( x ) affect f ( g ( x ) ) f(g(x)) 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 ) s(t) s ( t ) , velocity v ( t ) = s ′ ( t ) v(t) = s'(t) v ( t ) = s ′ ( t ) , acceleration a ( t ) = v ′ ( t ) a(t) = v'(t) a ( t ) = v ′ ( t )
Use chain rule if s ( t ) s(t) s ( t ) is composite function
Marginal cost and revenue in economics:
Cost C ( x ) C(x) C ( x ) , revenue R ( x ) R(x) R ( x ) , use chain rule if C ( x ) C(x) C ( x ) or R ( x ) R(x) 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