Understanding the Chain Rule
The chain rule states that if you have a composite function, say \(f(g(x))\), the derivative of this function can be expressed as:
\[
\frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x)
\]
This means that to differentiate a composite function, you first differentiate the outer function \(f\) evaluated at the inner function \(g(x)\), and then multiply it by the derivative of the inner function \(g(x)\).
When to Use the Chain Rule
You should use the chain rule when:
- You are differentiating a function that is composed of another function.
- The function involves powers, roots, trigonometric, exponential, or logarithmic functions nested within each other.
Common Chain Rule Questions and Answers
Below are some frequently asked questions regarding the chain rule, along with detailed answers to enhance your understanding.
Question 1: How do you differentiate \(h(x) = (3x + 2)^4\)?
Answer:
To differentiate \(h(x) = (3x + 2)^4\) using the chain rule:
1. Identify the outer function \(f(u) = u^4\) and the inner function \(g(x) = 3x + 2\).
2. Differentiate the outer function: \(f'(u) = 4u^3\).
3. Differentiate the inner function: \(g'(x) = 3\).
4. Apply the chain rule:
\[
h'(x) = f'(g(x)) \cdot g'(x) = 4(3x + 2)^3 \cdot 3 = 12(3x + 2)^3
\]
Thus, the derivative \(h'(x) = 12(3x + 2)^3\).
Question 2: What is the derivative of \(f(x) = \sin(5x^2)\)?
Answer:
To find the derivative of \(f(x) = \sin(5x^2)\):
1. Identify the outer function \(f(u) = \sin(u)\) and the inner function \(g(x) = 5x^2\).
2. Differentiate the outer function: \(f'(u) = \cos(u)\).
3. Differentiate the inner function: \(g'(x) = 10x\).
4. Apply the chain rule:
\[
f'(x) = f'(g(x)) \cdot g'(x) = \cos(5x^2) \cdot 10x = 10x \cos(5x^2)
\]
Therefore, the derivative \(f'(x) = 10x \cos(5x^2)\).
Question 3: Can you explain how to use the chain rule with logarithmic functions? For example, find the derivative of \(f(x) = \ln(2x + 1)\).
Answer:
To differentiate \(f(x) = \ln(2x + 1)\):
1. Identify the outer function \(f(u) = \ln(u)\) and the inner function \(g(x) = 2x + 1\).
2. Differentiate the outer function: \(f'(u) = \frac{1}{u}\).
3. Differentiate the inner function: \(g'(x) = 2\).
4. Apply the chain rule:
\[
f'(x) = f'(g(x)) \cdot g'(x) = \frac{1}{2x + 1} \cdot 2 = \frac{2}{2x + 1}
\]
Thus, the derivative \(f'(x) = \frac{2}{2x + 1}\).
Question 4: How do you differentiate a function that involves multiple layers, such as \(f(x) = e^{(3x^2 + 2)}\)?
Answer:
To differentiate \(f(x) = e^{(3x^2 + 2)}\):
1. Identify the outer function \(f(u) = e^u\) and the inner function \(g(x) = 3x^2 + 2\).
2. Differentiate the outer function: \(f'(u) = e^u\).
3. Differentiate the inner function: \(g'(x) = 6x\).
4. Apply the chain rule:
\[
f'(x) = f'(g(x)) \cdot g'(x) = e^{(3x^2 + 2)} \cdot 6x
\]
Therefore, the derivative \(f'(x) = 6x e^{(3x^2 + 2)}\).
Practice Problems for Chain Rule Mastery
To solidify your understanding of the chain rule, try solving the following problems:
- Find the derivative of \(f(x) = \tan(4x + 1)\).
- Differentiate \(g(x) = (x^2 + 3)^5\).
- Compute the derivative of \(h(x) = \sqrt{7x^3 + 5x}\).
- Determine the derivative of \(j(x) = \cos(3x^2 - 4x)\).
- Find \(k(x) = \ln(5x^2 + 3x + 1)\).
Conclusion
In this article, we have explored various chain rule questions and answers that illustrate how to apply this essential mathematical principle. Mastering the chain rule is vital for anyone studying calculus, as it enables you to tackle complex derivatives with confidence. By practicing the provided problems and reviewing the explanations, you will gain a deeper understanding of the chain rule and be better prepared for more advanced topics in calculus.
Frequently Asked Questions
What is the chain rule in calculus?
The chain rule is a fundamental theorem in calculus that describes how to differentiate composite functions. It states that if you have two functions, f(g(x)), the derivative is f'(g(x)) g'(x).
How do you apply the chain rule to find the derivative of sin(3x^2)?
To differentiate sin(3x^2), identify the outer function as sin(u) and the inner function as u = 3x^2. Using the chain rule, the derivative is cos(3x^2) (6x) = 6x cos(3x^2).
Can you provide an example of a chain rule problem involving exponential functions?
Sure! For the function e^(2x^3), let u = 2x^3. The derivative using the chain rule is e^(2x^3) (6x^2) = 6x^2 e^(2x^3).
What is the importance of the chain rule in real-world applications?
The chain rule is crucial in various fields such as physics, engineering, and economics, as it helps in modeling rates of change in complex systems where one variable depends on another.
How do you use the chain rule with trigonometric and logarithmic functions?
For example, to differentiate ln(cos(x)), set u = cos(x). The derivative is (1/cos(x)) (-sin(x)) = -tan(x). This shows the chain rule in action with both logarithmic and trigonometric functions.
What common mistakes should one avoid when using the chain rule?
Common mistakes include forgetting to multiply by the derivative of the inner function, misidentifying the outer and inner functions, and applying the rule incorrectly to non-composite functions.
