Glossary

7.2 Derivatives of composite functions

2 min read•july 22, 2024

The is a powerful tool for differentiating composite functions. It allows us to break down complex functions into simpler parts, making differentiation more manageable. This technique is crucial for solving real-world problems involving rates of change.

Mastering the chain rule opens up a world of possibilities in calculus. From to exponentials and logarithms, this rule helps us tackle a wide range of composite functions. It's a key skill for any calculus student to develop.

Derivatives of Composite Functions

Chain rule for composite functions

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  • Differentiates composite functions (function of a function)
  • If f(x)=h(g(x))f(x) = h(g(x)), then f′(x)=h′(g(x))⋅g′(x)f'(x) = h'(g(x)) \cdot g'(x)
    • Identifies outer function h(x)h(x) and inner function g(x)g(x)
    • Differentiates outer function, keeping inner function as a variable
    • Multiplies result by derivative of inner function
  • Examples:
    • If f(x)=(3x2+1)5f(x) = (3x^2 + 1)^5, then f′(x)=5(3x2+1)4⋅6xf'(x) = 5(3x^2 + 1)^4 \cdot 6x
    • If f(x)=2x−1f(x) = \sqrt{2x - 1}, then f′(x)=122x−1⋅2f'(x) = \frac{1}{2\sqrt{2x - 1}} \cdot 2

Order in chain rule application

  • Applies chain rule to complex composite functions by working from outside in
    • Differentiates outermost function first, keeping inner functions as variables
    • Multiplies result by derivative of next inner function
    • Continues process until all functions differentiated
  • Example: If f(x)=sin⁡(ex2+1)f(x) = \sin(e^{x^2 + 1}), then:
    1. f′(x)=cos⁡(ex2+1)⋅ddx(ex2+1)f'(x) = \cos(e^{x^2 + 1}) \cdot \frac{d}{dx}(e^{x^2 + 1})
    2. f′(x)=cos⁡(ex2+1)⋅ex2+1⋅ddx(x2+1)f'(x) = \cos(e^{x^2 + 1}) \cdot e^{x^2 + 1} \cdot \frac{d}{dx}(x^2 + 1)
    3. f′(x)=cos⁡(ex2+1)⋅ex2+1⋅2xf'(x) = \cos(e^{x^2 + 1}) \cdot e^{x^2 + 1} \cdot 2x

Derivatives of complex composite functions

  • Trigonometric functions:
    • If f(x)=sin⁡(g(x))f(x) = \sin(g(x)), then f′(x)=cos⁡(g(x))⋅g′(x)f'(x) = \cos(g(x)) \cdot g'(x)
    • If f(x)=cos⁡(g(x))f(x) = \cos(g(x)), then f′(x)=−sin⁡(g(x))⋅g′(x)f'(x) = -\sin(g(x)) \cdot g'(x)
    • If f(x)=tan⁡(g(x))f(x) = \tan(g(x)), then f′(x)=sec⁡2(g(x))⋅g′(x)f'(x) = \sec^2(g(x)) \cdot g'(x)
  • :
    • If f(x)=eg(x)f(x) = e^{g(x)}, then f′(x)=eg(x)⋅g′(x)f'(x) = e^{g(x)} \cdot g'(x)
    • If f(x)=ag(x)f(x) = a^{g(x)} (a>0a > 0, a≠1a \neq 1), then f′(x)=ag(x)⋅ln⁡(a)⋅g′(x)f'(x) = a^{g(x)} \cdot \ln(a) \cdot g'(x)
  • :
    • If f(x)=ln⁡(g(x))f(x) = \ln(g(x)), then f′(x)=1g(x)⋅g′(x)f'(x) = \frac{1}{g(x)} \cdot g'(x)
    • If f(x)=log⁡a(g(x))f(x) = \log_a(g(x)) (a>0a > 0, a≠1a \neq 1), then f′(x)=1g(x)⋅ln⁡(a)⋅g′(x)f'(x) = \frac{1}{g(x) \cdot \ln(a)} \cdot g'(x)

Chain rule in real-world applications

  • Finds rates of change in real-world problems using composite functions
  • Steps to solve:
    1. Identifies relating quantities in problem
    2. Uses chain rule to differentiate composite function
    3. Substitutes given values into derivative to find rate of change
  • Example: Volume of sphere increasing at 10 cm³/min, find rate radius increasing when radius is 5 cm
    • Volume of sphere: V=43πr3V = \frac{4}{3}\pi r^3
    • dVdt=4πr2⋅drdt\frac{dV}{dt} = 4\pi r^2 \cdot \frac{dr}{dt}
    • Substitute values: 10=4π(5)2⋅drdt10 = 4\pi (5)^2 \cdot \frac{dr}{dt}
    • Solve for drdt\frac{dr}{dt} to find rate of change of radius

Key Terms to Review (13)

(f(g(x)))' = f'(g(x)) * g'(x): This expression represents the derivative of a composite function, known as the chain rule in calculus. It shows how to differentiate a function that is composed of another function, illustrating that to find the overall rate of change, you need to multiply the derivative of the outer function evaluated at the inner function by the derivative of the inner function. The chain rule is a foundational concept in calculus, allowing for the analysis of more complex relationships between variables.
Chain Rule: The chain rule is a fundamental technique in calculus used to differentiate composite functions, allowing us to find the derivative of a function that is made up of other functions. This rule is crucial for understanding how different rates of change are interconnected and enables us to tackle complex differentiation problems involving multiple layers of functions.
Composite Function: A composite function is formed when one function is applied to the result of another function. This means that you take the output of one function and use it as the input for another, creating a new function altogether. Composite functions allow us to understand how two functions interact and combine, which is crucial for understanding relationships in mathematics.
Dy/dx: The notation $$\frac{dy}{dx}$$ represents the derivative of a function, indicating the rate at which the dependent variable $$y$$ changes with respect to the independent variable $$x$$. This concept is essential for understanding how functions behave and helps in solving problems related to tangents, slopes, and rates of change. The derivative encapsulates the instantaneous rate of change, allowing for the analysis of motion and the dynamics of systems.
Exponential Functions: Exponential functions are mathematical functions of the form $$f(x) = a imes b^{x}$$, where $$a$$ is a constant, $$b$$ is a positive real number, and $$x$$ is the exponent. They describe processes that grow or decay at a constant rate proportional to their current value, making them crucial in modeling real-world phenomena such as population growth and radioactive decay.
F(g(x)): The expression f(g(x)) represents the composition of two functions, where g(x) is applied first and then the result is used as the input for the function f. This concept showcases how functions can be combined to create new functions, allowing for complex relationships and transformations. Understanding this composition is crucial for unraveling the behavior of combined functions and calculating their derivatives efficiently.
Find the derivative of sin(x^2): Finding the derivative of sin(x^2) involves using the chain rule to differentiate a composite function, where sin is the outer function and x^2 is the inner function. The chain rule states that the derivative of a composite function can be computed by multiplying the derivative of the outer function evaluated at the inner function by the derivative of the inner function. This process highlights how calculus allows us to analyze how changes in one variable affect another in a composition of functions.
Fundamental theorem of calculus: The fundamental theorem of calculus establishes the connection between differentiation and integration, showing that they are essentially inverse processes. This theorem consists of two parts: the first part guarantees that if a function is continuous on a closed interval, then its definite integral can be computed using its antiderivative. The second part states that the derivative of the integral of a function is equal to the original function. This fundamental concept links the concepts of derivatives and antiderivatives together, playing a crucial role in understanding how these processes relate to one another.
Higher-order derivatives: Higher-order derivatives are the derivatives of a function taken more than once. They provide insights into the behavior of a function by showing how its rate of change itself changes, offering deeper understanding of its curvature and concavity. Higher-order derivatives play a critical role in various mathematical applications, including optimization and curve sketching, and are especially relevant when using rules such as the product rule and when dealing with composite functions.
Logarithmic Functions: Logarithmic functions are the inverses of exponential functions, expressing the power to which a base must be raised to produce a given number. They are essential in various applications, such as simplifying complex calculations and modeling real-world phenomena, like growth and decay. The properties of logarithms, including their relationship with exponentials, play a crucial role in differentiation and solving equations involving rates.
Mean Value Theorem: The Mean Value Theorem states that for a function that is continuous on a closed interval and differentiable on the open interval, there exists at least one point where the derivative equals the average rate of change of the function over that interval. This theorem provides a bridge between the behavior of a function and its derivatives, showing how slopes relate to overall changes.
Second Derivative: The second derivative is the derivative of the derivative of a function, providing insight into the function's rate of change in relation to its own rate of change. This concept helps us understand not just how a function is changing, but also how the rate of that change is itself changing, revealing key features like concavity and potential inflection points. Additionally, the second derivative plays a significant role in analyzing the behavior of composite functions and the implications of implicit differentiation.
Trigonometric Functions: Trigonometric functions are mathematical functions that relate the angles of a triangle to the lengths of its sides, commonly used in various fields like physics, engineering, and computer science. These functions include sine, cosine, tangent, cosecant, secant, and cotangent, and they play a vital role in understanding periodic phenomena. Their properties are essential when discussing derivatives, continuity, and the application of rules in calculus.
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