Chain Rule

Definition

The chain rule is a fundamental concept in calculus that allows for the differentiation of composite functions. It provides a systematic way to find the derivative of a function that is composed of other functions.

5 Must Know Facts For Your Next Test

1. The chain rule is used to find the derivative of a composite function, where the inner function is a function of the outer function's variable.
2. The chain rule is essential in the Fundamental Theorem of Calculus, as it allows for the differentiation of the integrand in the second part of the theorem.
3. The chain rule is crucial in the context of substitution, as it enables the differentiation of the new variable introduced during the substitution process.
4. The chain rule is used to differentiate functions involving exponential and logarithmic functions, as well as inverse trigonometric functions.
5. The chain rule is also applied in the calculus of parametric curves, where the derivatives of the parametric equations are found using the chain rule.

Review Questions

• Explain how the chain rule is used in the context of the Fundamental Theorem of Calculus.
  • The chain rule is essential in the Fundamental Theorem of Calculus, as it allows for the differentiation of the integrand in the second part of the theorem. When evaluating the integral $\int f(g(x)) g'(x) dx$, the chain rule is used to find the derivative of the composite function $f(g(x))$, which is then integrated to obtain the original function. This connection between differentiation and integration is a key aspect of the Fundamental Theorem of Calculus, and the chain rule is a crucial tool in this process.
• Describe how the chain rule is applied in the context of substitution.
  • The chain rule is crucial in the context of substitution, as it enables the differentiation of the new variable introduced during the substitution process. When performing a substitution, such as $u = g(x)$, the chain rule is used to find the derivative $\frac{dy}{dx}$ in terms of the new variable $u$ and its derivative $\frac{du}{dx}$. This allows for the integration of the original function to be simplified by working with the new variable, while still maintaining the correct derivative. The chain rule ensures that the differentiation and integration steps are properly connected during the substitution process.
• Analyze how the chain rule is used in the differentiation of functions involving exponential, logarithmic, and inverse trigonometric functions.
  • The chain rule is used extensively in the differentiation of functions involving exponential, logarithmic, and inverse trigonometric functions. For example, when differentiating a function like $f(x) = e^{g(x)}$, the chain rule is applied to find $f'(x) = e^{g(x)} \cdot g'(x)$. Similarly, for a function like $f(x) = \log(g(x))$, the chain rule is used to obtain $f'(x) = \frac{g'(x)}{g(x)}$. The same principle applies to functions involving inverse trigonometric functions, such as $f(x) = \sin^{-1}(g(x))$, where the chain rule is used to find the derivative $f'(x) = \frac{g'(x)}{\sqrt{1 - (g(x))^2}}$. The chain rule is a crucial tool in the differentiation of these types of composite functions.

Related terms

Composite Function: A composite function is a function that is formed by combining two or more functions, where the output of one function becomes the input of the next function.

Derivative:

The derivative of a function represents the rate of change of the function at a specific point, and it is a fundamental concept in calculus.

Substitution:

Substitution is a technique used in calculus to simplify the integration of a function by replacing the original variable with a new variable that makes the integration easier.