Question

Explain what the chain rule and product rules in calculus are.

Answer 1

The chain rule and product rule are fundamental differentiation rules in calculus that help us compute the derivatives of composite functions and products of functions, respectively. Both rules are essential techniques when finding the derivatives of more complex expressions.

1. Chain Rule:
The chain rule is used to differentiate composite functions, i.e., functions that involve one function being applied within another function. Given a composite function \(f(x)=\) \(g(h(x))\), the chain rule states that the derivative of the composite function with respect to \(x\) is the product of the derivative of the outer function with respect to the inner function and the derivative of the inner function with respect to \(x\).
Mathematically, the chain rule is expressed as:
\[
f^{\prime}(x)=g^{\prime}(h(x)) \cdot h^{\prime}(x)
\]

 

Example:
Consider the function \(f(x)=\sin \left(x^2\right)\). Here, \(g(x)=\sin (x)\) and \(h(x)=x^2\). We want to find the derivative of \(f(x)\).
Using the chain rule:
\[
f^{\prime}(x)=g^{\prime}(h(x)) \cdot h^{\prime}(x)=\left(\cos \left(x^2\right)\right) \cdot(2 x)=2 x \cos \left(x^2\right)
\]
2. Product Rule:
The product rule is used to differentiate the product of two functions. Given a product of functions \(f(x)=g(x) \cdot h(x)\), the product rule states that the derivative of the product function with respect to \(x\) is the derivative of the first function times the second function plus the first function times the derivative of the second function.
Mathematically, the product rule is expressed as:
\[
f^{\prime}(x)=g^{\prime}(x) h(x)+g(x) h^{\prime}(x)
\]

Example:
Consider the function \(f(x)=x^3 \cdot e^x\). Here, \(g(x)=x^3\) and \(h(x)=e^x\). We want to find the derivative of \(f(x)\).
Using the product rule:
\[
f^{\prime}(x)=g^{\prime}(x) h(x)+g(x) h^{\prime}(x)=\left(3 x^2\right)\left(e^x\right)+\left(x^3\right)\left(e^x\right)=e^x\left(3 x^2+x^3\right)
\]
By applying the chain rule and product rule, you can differentiate various functions that involve compositions or products of other functions. These rules are essential when dealing with more complex expressions in calculus.