Chain Rule in Derivatives: Step-by-Step Guide for Mastering Calculus

The Chain Rule in Derivatives is a fundamental technique in calculus used to differentiate composite functions, where one function is nested inside another. It simplifies the process by breaking down the derivative of the outer function and multiplying it by the derivative of the inner function. The chain rule is widely applied in fields like physics, economics, and engineering to calculate rates of change in interconnected systems.

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1. Introduction to the Chain Rule in Derivatives:

In calculus, the chain rule is one of the most crucial techniques for finding the derivative of composite functions. It enables us to differentiate complex functions by breaking them down into simpler parts. Whether dealing with a single variable's functions or more complex multivariable functions, understanding the chain rule is essential for mastering differentiation. In this blog, we will dive into the fundamentals of the chain rule, explore its formulas, and provide solved examples to ensure a clear understanding of this key concept.

2. What is Chain Rule in Derivatives:

The chain rule is a formula for computing the derivative of a composite function. A composite function occurs when one function is nested inside another, like $f(g(x))$, where $f$ is applied to $g(x)$. The chain rule helps us differentiate such functions by systematically breaking them down.

Chain Rule Steps:

1. Identify the outer function $f$ and the inner function $g$.
2. Differentiate the outer function $f$ with respect to the inner function $g(x)$.
3. Multiply the result by the derivative of the inner function $g(x)$.

For example, to find the derivative of $f(g(x))$, the chain rule is expressed as:

$\dfrac{d}{dx} \left[ f(g(x)) \right] = f'(g(x)) \cdot g'(x)$

This formula essentially states that we first differentiate the outer function $f$ while treating $g(x)$ as a constant, then multiply by the derivative of the inner function $g(x)$.

3. Chain Rule in Derivatives Formula:

The chain rule can be applied in different forms based on the functions' presentation. Below are two common variations of the chain rule formula.

Chain Rule in Derivatives Formula 1:

For a composite function $f(g(x))$, the chain rule is given by:

$\dfrac{d}{dx} \left[ f(g(x)) \right] = f'(g(x)) \cdot g'(x)$

Here, $f$ is the outer function, and $g(x)$ is the inner function.

Chain Rule in Derivatives Formula 2:

In the case of multivariable functions, where $z = f(x, y)$ and $x = g(t)$, $y = h(t)$, the chain rule takes the form:

$\dfrac{dz}{dt} = \dfrac{\partial z}{\partial x} \cdot \dfrac{dx}{dt} \dfrac{\partial z}{\partial y} \cdot \dfrac{dy}{dt}$

This formula accounts for changes in both $x$ and $y$ as functions of $t$.

4. Double Chain Rule in Derivatives:

The double chain rule is used when dealing with functions that involve two layers of composition. For example, consider a function $f(g(h(x)))$, which has three nested functions. The double chain rule allows us to differentiate such functions step by step:

$\dfrac{d}{dx} [f(g(h(x)))] = f'(g(h(x))) \cdot g'(h(x)) \cdot h'(x)$

Each step involves applying the chain rule to differentiate the outer function, then the next inner function, and so on.

5. Chain Rule in Derivatives Solved Examples:

Question: 1.

Find the derivative of $f(x) = \cos(3x)$ using the chain rule.

Solution:

Step 1: Identify the outer and inner functions.

• Outer function: $f(u) = \cos(u)$
• Inner function: $u(x) = 3x$

Step 2: Differentiate the outer function. $\dfrac{d}{du} \cos(u) = -\sin(u)$

Step 3: Differentiate the inner function. $\dfrac{d}{dx} (3x) = 3$

Step 4: Apply the chain rule. $\dfrac{d}{dx} \cos(3x) = -\sin(3x) \cdot 3$

Final Answer: $\dfrac{d}{dx} \cos(3x) = -3 \sin(3x)$

Question: 2.

Find the derivative of $f(x) = (2x^2 1)^5$ using the chain rule.

Solution:

Step 1: Identify the outer and inner functions.

• Outer function: $f(u) = u^5$
• Inner function: $u(x) = 2x^2 1$

Step 2: Differentiate the outer function. $\dfrac{d}{du} (u^5) = 5u^4$

Step 3: Differentiate the inner function. $\dfrac{d}{dx} (2x^2 1) = 4x$

Step 4: Apply the chain rule. $\dfrac{d}{dx} (2x^2 1)^5 = 5(2x^2 1)^4 \cdot 4x$

Final Answer: $\dfrac{d}{dx} (2x^2 1)^5 = 20x(2x^2 1)^4$

Question: 3.

Find the derivative of $f(x) = \ln(5x 3)$ using the chain rule.

Solution:

Step 1: Identify the outer and inner functions.

• Outer function: $f(u) = \ln(u)$
• Inner function: $u(x) = 5x 3$

Step 2: Differentiate the outer function. $\dfrac{d}{du} \ln(u) = \dfrac{1}{u}$

Step 3: Differentiate the inner function. $\dfrac{d}{dx} (5x 3) = 5$

Step 4: Apply the chain rule. $\dfrac{d}{dx} \ln(5x 3) = \dfrac{1}{5x 3} \cdot 5$

Final Answer: $\dfrac{d}{dx} \ln(5x 3) = \dfrac{5}{5x 3}$

Question: 4.

Find the derivative of $f(x) = e^{x^3}$ using the chain rule.

Solution:

Step 1: Identify the outer and inner functions.

• Outer function: $f(u) = e^u$
• Inner function: $u(x) = x^3$

Step 2: Differentiate the outer function. $\dfrac{d}{du} e^u = e^u$

Step 3: Differentiate the inner function. $\dfrac{d}{dx} (x^3) = 3x^2$

Step 4: Apply the chain rule. $\dfrac{d}{dx} e^{x^3} = e^{x^3} \cdot 3x^2$

Final Answer: $\dfrac{d}{dx} e^{x^3} = 3x^2 e^{x^3}$

Question: 5.

Find the derivative of $f(x) = \sin(\ln(x))$ using the chain rule.

Solution:

Step 1: Identify the outer and inner functions.

• Outer function: $f(u) = \sin(u)$
• Inner function: $u(x) = \ln(x)$

Step 2: Differentiate the outer function. $\dfrac{d}{du} \sin(u) = \cos(u)$

Step 3: Differentiate the inner function. $\dfrac{d}{dx} \ln(x) = \dfrac{1}{x}$

Step 4: Apply the chain rule. $\dfrac{d}{dx} \sin(\ln(x)) = \cos(\ln(x)) \cdot \dfrac{1}{x}$

Final Answer: $\dfrac{d}{dx} \sin(\ln(x)) = \dfrac{\cos(\ln(x))}{x}$

6. Practice Questions on Chain Rule in Derivatives:

Q:1. Find the derivative of $f(x) = \ln(x^2 1)$.

Q:2. Differentiate $f(x) = \cos(5x^3)$.

Q:3. Use the chain rule to differentiate $f(x) = (3x 2)^4$.

Q:4. Differentiate $f(x) = e^{\sin(x)}$.

Q:5. Find the derivative of $f(x) = \sqrt{1 2x^3}$.

7. FAQs on Chain Rule in Derivatives:

What is the Chain Rule in derivatives?

The Chain Rule is a formula in calculus used to differentiate composite functions. It states that the derivative of a composite function $f(g(x))$ is the derivative of the outer function evaluated at the inner function, multiplied by the derivative of the inner function.

When should I use the Chain Rule?

You should use the Chain Rule whenever you are differentiating a function that is the composition of two or more functions, such as $f(g(x))$, where $g(x)$ is inside $f(x)$.

What is the formula for the Chain Rule?

The Chain Rule formula is:$\dfrac{d}{dx} \left[ f(g(x)) \right] = f'(g(x)) \cdot g'(x)$ It states that you differentiate the outer function $f$ and multiply it by the derivative of the inner function $g(x)$.

Can the Chain Rule be applied more than once?

Yes, when differentiating functions with multiple nested layers, like $f(g(h(x)))$, the Chain Rule can be applied repeatedly. This is known as the Double Chain Rule or Multiple Chain Rule.

How do I know which function is the inner and which is the outer?

The inner function is the one that is "inside" another function, typically represented as $g(x)$, while the outer function is the one applied to the inner function, represented as $f(x)$. For example, in $\sin(x^2)$, $x^2$ is the inner function, and $\sin(u)$ is the outer function.

Can the Chain Rule be used for multivariable functions?

Yes, the Chain Rule extends to multivariable functions, allowing you to differentiate functions where multiple variables depend on a single parameter. This version is often called the Multivariable Chain Rule.

What is the Double Chain Rule?

The Double Chain Rule is used for differentiating functions that involve two layers of composition, such as $f(g(h(x)))$. It involves applying the Chain Rule twice to work through each layer of the function.

Is the Chain Rule important in real-world applications?

Absolutely. The Chain Rule is crucial in fields like physics, engineering, and economics, where many processes involve nested functions. For example, it's used to compute rates of change in systems where variables depend on each other, such as in dynamic systems and optimization problems.

8. Real-life Application of Chain Rule in Derivatives:

The chain rule is widely used in fields like physics, economics, and engineering. For example:

• In physics, the chain rule helps compute rates of change in dynamic systems, such as velocity and acceleration when time and space are connected.

• In economics, it’s used to model how changes in one variable, like interest rates, affect other dependent variables like investment.

• In engineering, the chain rule is applied in optimization problems, where functions with multiple layers of dependencies need to be differentiated to find optimal solutions.

9. Conclusion:

The chain rule in derivatives is a powerful tool for differentiating composite functions. Whether you're dealing with simple compositions or more complex multivariable functions, mastering the chain rule enables you to break down difficult problems into manageable parts. With its wide application in both theoretical and practical problems, from mathematics to physics and engineering, the chain rule is an essential technique that every student and professional should be comfortable using.

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