The Chain Rule

The Chain Rule

The Chain Rule is a fundamental technique in calculus that allows us to differentiate composite functions. A composite function is one in which one function is nested inside another. The Chain Rule helps us find the derivative of such functions by taking the derivative of the outer function and multiplying it by the derivative of the inner function.

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The Chain Rule Formula

If you have a composite function $f(x) = g(h(x))$, where $g$ is a function of $h(x)$, and $h(x)$ is a function of $x$, the Chain Rule states that the derivative of $f(x)$ is:

$$f'(x) = g'(h(x)) \cdot h'(x)$$

This means you:

1. Differentiate the outer function $g$ with respect to its argument $h(x)$.
2. Multiply by the derivative of the inner function $h(x)$.

Steps to Apply the Chain Rule:

1. Identify the outer function $g(x)$ and the inner function $h(x)$.
2. Differentiate the outer function $g(x)$ with respect to $h(x)$, leaving the inner function $h(x)$ unchanged.
3. Multiply this derivative by the derivative of the inner function $h(x)$.

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Example 1: Differentiating a Simple Composite Function

Let’s differentiate $f(x) = \sin(x^2)$.

1. Identify the outer and inner functions:

   • Outer function $g(u) = \sin(u)$, where $u = x^2$.
   • Inner function $h(x) = x^2$.

2. Differentiate the outer function with respect to $u$:

   $$\frac{d}{du}(\sin(u)) = \cos(u)$$

3. Differentiate the inner function with respect to $x$:

   $$\frac{d}{dx}(x^2) = 2x$$

4. Apply the Chain Rule:

   $$f'(x) = \cos(x^2) \cdot 2x$$

So, the derivative of $f(x) = \sin(x^2)$ is:

$$f'(x) = 2x \cos(x^2)$$

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Example 2: Differentiating a More Complex Function

Let’s differentiate $f(x) = \ln(3x^2 + 5)$.

1. Identify the outer and inner functions:

   • Outer function $g(u) = \ln(u)$, where $u = 3x^2 + 5$.
   • Inner function $h(x) = 3x^2 + 5$.

2. Differentiate the outer function with respect to $u$:

   $$\frac{d}{du}(\ln(u)) = \frac{1}{u}$$

3. Differentiate the inner function with respect to $x$:

   $$\frac{d}{dx}(3x^2 + 5) = 6x$$

4. Apply the Chain Rule:

   $$f'(x) = \frac{1}{3x^2 + 5} \cdot 6x$$

So, the derivative of $f(x) = \ln(3x^2 + 5)$ is:

$$f'(x) = \frac{6x}{3x^2 + 5}$$

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Example 3: Chain Rule with Trigonometric Functions

Let’s differentiate $f(x) = \sin(5x^3)$.

1. Identify the outer and inner functions:

   • Outer function $g(u) = \sin(u)$, where $u = 5x^3$.
   • Inner function $h(x) = 5x^3$.

2. Differentiate the outer function with respect to $u$:

   $$\frac{d}{du}(\sin(u)) = \cos(u)$$

3. Differentiate the inner function with respect to $x$:

   $$\frac{d}{dx}(5x^3) = 15x^2$$

4. Apply the Chain Rule:

   $$f'(x) = \cos(5x^3) \cdot 15x^2$$

So, the derivative of $f(x) = \sin(5x^3)$ is:

$$f'(x) = 15x^2 \cos(5x^3)$$

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Example 4: Chain Rule with Exponential Functions

Let’s differentiate $f(x) = e^{3x^2 + 2x}$.

1. Identify the outer and inner functions:

   • Outer function $g(u) = e^u$, where $u = 3x^2 + 2x$.
   • Inner function $h(x) = 3x^2 + 2x$.

2. Differentiate the outer function with respect to $u$:

   $$\frac{d}{du}(e^u) = e^u$$

3. Differentiate the inner function with respect to $x$:

   $$\frac{d}{dx}(3x^2 + 2x) = 6x + 2$$

4. Apply the Chain Rule:

   $$f'(x) = e^{3x^2 + 2x} \cdot (6x + 2)$$

So, the derivative of $f(x) = e^{3x^2 + 2x}$ is:

$$f'(x) = (6x + 2)e^{3x^2 + 2x}$$

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Key Points to Remember

• The Chain Rule is used when differentiating a composite function, which is a function inside another function.
• The Chain Rule formula is: $f'(x) = g'(h(x)) \cdot h'(x)$, where $g(h(x))$ is the composite function.
• Step-by-step approach:
  1. Differentiate the outer function with respect to the inner function.
  2. Multiply by the derivative of the inner function.
• Be mindful of nested functions, including polynomials, trigonometric functions, logarithms, and exponentials.

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Summary of the Chain Rule:

1. Composite Function: $f(x) = g(h(x))$
2. Derivative: $f'(x) = g'(h(x)) \cdot h'(x)$
3. Steps:
   • Differentiate the outer function.
   • Multiply by the derivative of the inner function.
4. Application: The Chain Rule applies to a wide range of composite functions (including polynomials, trigonometric, logarithmic, and exponential functions).

By mastering the Chain Rule, you can differentiate complex functions that involve multiple layers of composition and relationships between functions.