The chain rule is a fundamental concept in calculus used for differentiating composite functions. Let's break it down step by step:
1. What is a composite function?
A composite function is a function of a function. For example, if we have f(x) = x² and g(x) = sin(x), then f(g(x)) = sin²(x) is a composite function.
2. The Chain Rule Formula:
For a composite function y = f(g(x)), the derivative is:
dxdy=f′(g(x))⋅g′(x) 3. How it works:
- f′(g(x)) is the derivative of the outer function
- g′(x) is the derivative of the inner function
4. Step-by-step Example:
Let's differentiate y = (x² + 1)³
Step 1: Identify the composite parts
- Outer function: f(u) = u³
- Inner function: g(x) = x² + 1
Step 2: Find the derivatives of each part
- f'(u) = 3u²
- g'(x) = 2x
Step 3: Apply the chain rule
dxdy=f′(g(x))⋅g′(x)=3(x2+1)2⋅2x=6x(x2+1)2 5. Another Example:
Let's differentiate y = sin(x²)
Step 1: Identify the composite parts
- Outer function: f(u) = sin(u)
- Inner function: g(x) = x²
Step 2: Find the derivatives of each part
- f'(u) = cos(u)
- g'(x) = 2x
Step 3: Apply the chain rule
dxdy=f′(g(x))⋅g′(x)=cos(x2)⋅2x=2xcos(x2) 6. Graphical Representation:
Here's a graph showing the original function y = (x² + 1)³ and its derivative:
The blue curve represents the original function, while the red curve shows its derivative. Notice how the derivative's value corresponds to the slope of the tangent line at each point of the original function.
The chain rule allows us to break down complex, nested functions into simpler parts, making differentiation more manageable. It's a powerful tool in calculus with applications in physics, economics, and many other fields.
Would you like me to provide more examples or clarify any part of this explanation?