Introduction to Chain Rule

The derivative is a fundamental concept of calculus that involves the rate of the instantaneous change in a function. There are some derivative rules to calculate the rate of change of different functions like exponential, trigonometric, or logarithmic functions, etc. Sometimes, we have to deal with a combination of two functions. For this, the chain rule formula is used to find the derivative of two combined functions. Let’s understand how to apply the chain rule to find a derivative.

Understanding the Chain Rule

There are different rules of differentiation in calculus. All of these rules are important to find the rate of instantaneous change of a function. The chain rule is a rule of expressing derivative of a function which is a combination of two functions. It calculates the rate of change of a function in relation to the other function. 

By definition, the chain rule for a function $f(g(x))$ is stated as:

“The derivative of $f(g(x))$ is equal to the derivative of y with respect to u multiplied with the derivative of u with respect to x, where $y=f(u)$ and $u=g(x)$.”

The variable u is used to replace the second function so that it can be easily differentiated.

Chain Rule Formula

If two functions f(x) and g(x) are in a combination form such as f(x) is a function of g(x) i.e. $f(g(x)$. Then the chain rule formula is expressed as:

$\frac{d}{dx}[f(g(x))] = \frac{dy}{dx} = \frac{dy}{du} × \frac{du}{dx}$

Where, 

• $y=f(u)$ and $u = g(x)$
• $\frac{dy}{du}$ is the derivative of y with respect to u.
• $\frac{du}{dx}$ is the derivative of u with respect to x.

The chain rule can be used for three functions combined together.

Chain rule formula for three functions

If there are three functions f(x), g(x) and h(x) combined together. The chain rule can be used to calculate the derivative of such a function. The formula to calculate derivative of a combination of three functions is,

$\frac{dy}{dx} = \frac{dy}{du} × \frac{du}{dv} × \frac{dv}{dx}$

Where,

• $y=f(u)$, $u=g(v)$ and $v=h(x)$
• $\frac{dy}{du}$ is the derivative of y with respect to u.
• $\frac{du}{dv}$ is the derivative of u with respect to v.
• $\frac{dv}{dx}$ is the derivative of v with respect to x.

Or, it can be simply written as;

$\frac{dy}{dx} = f'(g(h(x))) \times g'(h(x)) \times h'(x)$

Where y=f(g(h(x))). Instead of using the product rule formula, the chain rule helps to find the derivative more easily.

How to apply chain rule formula?

The implementation of the chain rule of derivative is divided into a few steps. These steps assist us to calculate the derivative of two or more functions in fraction. These steps are:

1. Write the expression of the function. 
2. Identify the combination of two functions and name them as y=f(u) and u=g(x) functions.
3. Apply the derivative by using the chain rule formula. 
4. Calculate the derivative of both functions u and y one-by-one.
5. Multiply the derivative of y and u together as dy/du × du/dx.
6. Simplify if needed. 

Let’s understand the following examples by applying the chain rule of derivative.

Chain Rule Formula example 1

To calculate the derivative of e^x^3, we can use different techniques. The chain rule is one of the methods to evaluate derivative of e^x^3. 

$y = e^{x^3}$

In the above equation, $x^3$ can be replaced by a variable $u$. Therefore, 

$y = e^u$ and $u = x^3$

Now by using chain rule, we will calculate the derivative of both function y and u one-by-one.

$\frac{dy}{du} = e^u$ and $\frac{du}{dx} = 3x^2$

Since the chain rule is,

$\frac{dy}{dx} = \frac{dy}{du} × \frac{du}{dx}$

Now substituting the values of dy/du and du/dx.

$\frac{dy}{dx} = e^u × 3x^2$

Since $u = x^3$ then, 

$\frac{dy}{dx} =3x^2 e^{x^3}$

Since the given function is an exponential function then we can use the power rule of derivative also. 

Chain Rule Formula example 2

The derivative of cos 2x can be calculated by using the combination law of derivatives. So to calculate the derivative of cos 2x, we will use the following steps,

$y = \cos 2x$

In the above equation, the 2x can be replaced by another variable u. So, 

$y = \cos u$ and $u= 2x$

Now finding the derivative of both functions, 

$\frac{dy}{du} = -\cos u$ and $\frac{du}{dx} = 2$

Since the chain rule is,

$\frac{dy}{dx} = \frac{dy}{du} × \frac{du}{dx}$

Now substituting the values of dy/du and du/dx, we get, 

$\frac{dy}{dx} = - \cos u × 2$

Now using the value of u, we get

$\frac{dy}{dx} = -2\cos 2x$

Which is the derivative of cos 2x.

Applying Chain rule formula by using calculator

The derivative of a combination of two or more functions can be also calculated by using chain rule derivative calculator. It is an online tool that follows the chain rule derivative formula to find derivative. You can find it online by searching for a derivative calculator. For example, to calculate the derivative of cos x, the following steps are used by using this calculator.

1. Write the expression of the function in the input box such as, cos x.
2. Choose the variable to calculate the rate of change, which will be x in this example.
3. Review the input so that there will be no syntax error in the function. 
4. Now at the last step, click on the calculate button. By using this step, the chain rule calculator will provide the derivative of cos x quickly and accurately which will be $–sin x$.

Comparison between Chain rule and Quotient rule

The comparison between the chain and quotient rules can be easily analysed using the following difference table.

Conclusion

The chain rule is a rule of derivatives in calculus which is used to find the rate of change of a combination of two functions. It allows us to find the derivative by breaking down the function by using two variables. It has many applications in calculus as well as in real life.

Frequently Asked Questions

What is the chain rule in calculus?

The chain rule is a fundamental rule of differentiation in calculus that is used to find the derivative of a composite function, which is a combination of two or more functions. The chain rule formula involves taking the derivative of the outer function multiplied by the derivative of the inner function.

How do you apply the chain rule?

To apply the chain rule, you need to identify the inner and outer functions of the composite function and use the chain rule formula to find the derivative. The steps involved in applying the chain rule include writing the expression of the function, identifying the inner and outer functions, calculating the derivatives of each function, multiplying the derivatives, and simplifying the result.

Can the chain rule be used for three functions combined together?

Yes, the chain rule can be used for three functions combined together. The formula to calculate the derivative of a combination of three functions is dy/dx = dy/du × du/dv × dv/dx, where y=f(u), u=g(v), and v=h(x).

How do you calculate the derivative of e^x^3 using the chain rule?

To calculate the derivative of e^x^3 using the chain rule, we can replace x^3 with a variable u, where y=e^u and u=x^3. Then, we can calculate the derivative of both functions, dy/du = e^u and du/dx = 3x^2, and substitute them in the chain rule formula, dy/dx = dy/du × du/dx. Simplifying the result, we get dy/dx = 3x^2 e^x^3.

Is there an online tool to calculate the derivative using the chain rule?

Yes, there are several online tools available that can help calculate the derivative of a composite function using the chain rule. These tools use the chain rule formula to find the derivative and provide accurate results quickly and easily.