The chain rule is a fundamental concept in calculus used to find the derivative of a 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 for the next.

The Chain Rule Formula

Suppose you have two functions, $f(x)$ and $g(x)$, and you want to differentiate their composition $h(x) = f(g(x))$. The chain rule states that the derivative of $h(x)$ with respect to $x$ is:

$\frac{dh}{dx} = \frac{df}{dg} \cdot \frac{dg}{dx}$

Or, more commonly written as:

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

Explanation

1. Inner Function ($g(x)$): This is the function inside the composition. You first differentiate this function with respect to $x$.

2. Outer Function ($f(x)$): After differentiating the inner function, you substitute $g(x)$ into the derivative of the outer function, then multiply by the derivative of the inner function.

Example

Let's differentiate $h(x) = \sin(3x^2)$.

1. Identify the inner and outer functions:

   • Inner function: $g(x) = 3x^2$
   • Outer function: $f(u) = \sin(u)$, where $u = g(x) = 3x^2$.

2. Differentiate the inner function: $g'(x) = \frac{d}{dx}(3x^2) = 6x$

3. Differentiate the outer function: $f'(u) = \cos(u)$

4. Apply the chain rule: $h'(x) = \cos(3x^2) \cdot 6x$

So, the derivative of $h(x) = \sin(3x^2)$ is $h'(x) = 6x \cos(3x^2)$.

Would you like more details or have any questions?

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Related Questions:

1. How is the chain rule applied when differentiating implicit functions?
2. Can you explain how the chain rule works with trigonometric functions?
3. What are some common mistakes to avoid when using the chain rule?
4. How does the chain rule extend to higher dimensions, like in multivariable calculus?
5. How do you apply the chain rule to functions involving exponential and logarithmic functions?
6. How is the chain rule used in related rates problems?
7. Can you explain the proof of the chain rule?
8. What is the relationship between the chain rule and the power rule?

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Tip:

Always carefully identify the inner and outer functions when applying the chain rule, as this is key to correctly differentiating composite functions.