Chain Rule Rules

The chain rule is a fundamental concept in calculus used to differentiate composite functions. Understanding and applying the chain rule can simplify the process of finding derivatives of complex functions. Here’s a step-by-step guide to help you master the chain rule.

What is the Chain Rule?

The chain rule states that if you have two functions, $f$ and $g$, and you want to differentiate their composition, $(f\circ g)(x)=f(g(x))$, then the derivative of this composition is given by:

ddx[f(g(x))]=f′(g(x))⋅g′(x)\frac{d}{dx} [f(g(x))] = f'(g(x)) \cdot g'(x)dxd​[f(g(x))]=f′(g(x))⋅g′(x)

In simpler terms, you first differentiate the outer function $f$ evaluated at the inner function $g(x)$, and then multiply by the derivative of the inner function $g$.

Applying the Chain Rule

Let’s go through a detailed example to illustrate how to use the chain rule.

Example 1:

Find the derivative of h(x)=(3×2+2)5h(x) = (3x^2 + 2)^5h(x)=(3x2+2)5.

Step-by-Step Solution:

1. Identify the outer and inner functions:
   • Outer function: f(u)=u5f(u) = u^5f(u)=u5
   • Inner function: g(x)=3×2+2g(x) = 3x^2 + 2g(x)=3x2+2
2. Differentiate the outer function $f(u)$ with respect to $u$: f′(u)=5u4f'(u) = 5u^4f′(u)=5u4
3. Differentiate the inner function $g(x)$ with respect to $x$: g′(x)=6xg'(x) = 6xg′(x)=6x
4. Apply the chain rule: ddx[h(x)]=f′(g(x))⋅g′(x)\frac{d}{dx} [h(x)] = f'(g(x)) \cdot g'(x)dxd​[h(x)]=f′(g(x))⋅g′(x) Substitute g(x)=3×2+2g(x) = 3x^2 + 2g(x)=3x2+2 into f′(u)f'(u)f′(u): f′(g(x))=5(3×2+2)4f'(g(x)) = 5(3x^2 + 2)^4f′(g(x))=5(3x2+2)4 Then multiply by g′(x)g'(x)g′(x): ddx[(3×2+2)5]=5(3×2+2)4⋅6x\frac{d}{dx} [(3x^2 + 2)^5] = 5(3x^2 + 2)^4 \cdot 6xdxd​[(3x2+2)5]=5(3x2+2)4⋅6x Simplify the expression: ddx[(3×2+2)5]=30x(3×2+2)4\frac{d}{dx} [(3x^2 + 2)^5] = 30x(3x^2 + 2)^4dxd​[(3x2+2)5]=30x(3x2+2)4

Additional Examples

Example 2:

Find the derivative of $k(x)=\sin (4x)$.

1. Identify the outer and inner functions:
   • Outer function: $f(u)=\sin (u)$
   • Inner function: $g(x)=4x$
2. Differentiate the outer function $f(u)$ with respect to $u$: f′(u)=cos⁡(u)f'(u) = \cos(u)f′(u)=cos(u)
3. Differentiate the inner function $g(x)$ with respect to $x$: g′(x)=4g'(x) = 4g′(x)=4
4. Apply the chain rule: ddx[sin⁡(4x)]=cos⁡(4x)⋅4\frac{d}{dx} [\sin(4x)] = \cos(4x) \cdot 4dxd​[sin(4x)]=cos(4x)⋅4 Simplify the expression: ddx[sin⁡(4x)]=4cos⁡(4x)\frac{d}{dx} [\sin(4x)] = 4\cos(4x)dxd​[sin(4x)]=4cos(4x)

General Steps for Using the Chain Rule

1. Identify the composite function: Determine which function is inside another.
2. Differentiate the outer function: Treat the inner function as a single variable.
3. Differentiate the inner function: Find the derivative of the inner function with respect to its variable.
4. Multiply the derivatives: Combine the results by multiplying the derivative of the outer function (evaluated at the inner function) by the derivative of the inner function.

Tips and Tricks

• Practice: The chain rule becomes easier with practice. Work through various examples to build your understanding.
• Break it down: For complex functions, break them down into simpler parts and apply the chain rule step by step.
• Use parentheses: Clearly indicate the inner and outer functions using parentheses to avoid confusion.

Conclusion

Mastering the chain rule is crucial for tackling more advanced problems in calculus. By following the steps outlined above, you can confidently differentiate composite functions. If you need further assistance with calculus or any other mathematical concepts, consider using Translingua.ng for translation, transcription, and proofreading services. We also offer language classes and essay/statement writing services.

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