Introduction to Chain Rule
The Chain Rule is defined as a formula for computing the derivative of the composition of either two or more functions. For example, if ‘g’ and ‘f’ are functions, then the chain rule usually expresses the derivative of their composition.
Chain Rule Formula
The Chain Rule Formula is Represented as Follows:
\[\frac{dy}{dx}\] = \[\frac{dy}{du}\] . \[\frac{du}{dx}\]
Chain Rule Usage
The chain rule is used for differentiating a function, which is composed of the other functions in a specific way (it means, a composite function). By ‘composed’ it does not mean added, or multiplied, It means that you apply one function to the variable, then the other function (or a similar one) to the result and so on.
Let Us Take the Chain Rule Math Example.
For instance, (x2+3x+1)7 is given as a composite function. Here, the inner function is given as x2+3x+1, and the outer function is the power 7.
The derivative of the composite function is the individual functions’ product of the derivatives. Let us see how this works for the above function. Usually, we can start from the outside function. The power 7 derivative is 7 times the power 6, and so, the derivative is given as : 6(x2+3x+1)7
It is to be noted that, here, it is differentiated with respect to the variable that was inputted into the given function. Some would substitute u = x2+3x+1 and differentiate with respect to the value u.
Now, we have to multiply by the inner function derivative. The final result will be
\[\frac{d}{dx}\] (x2+3x+1)7 = 6(x2+3x+1)6 (2x+3)
We could also try replacing a term of the inner function by the composite function.
For Example,
\[\frac{d}{dx}\] (sin2x+3x+1)7= 6(sin2x+3x+1)6 (2 sin x cos x+3)
Where we have applied the chain rule to the sin2 function inside to the inner function.
Solved Examples
1. Differentiate f(x)=(1+x2)5
Solution:
Using the chain rule, let us solve the problem.
\[\frac{dy}{dx}\] = \[\frac{dy}{du}\] . \[\frac{du}{dx}\]
Let us Take
y = u5 and u = 1+x2
Then,
\[\frac{dy}{du}\] = \[\frac{d}{du}\](u5) = 5u4
\[\frac{dy}{du}\] = \[\frac{d}{dx}\](1+x2)=2x
\[\frac{dy}{dx}\] = 5u4.2x = 5(1+x2)4.2x
\[\frac{dy}{dx}\] =10x(1+x2)4
FAQs on Chain Rule Formula
1. What is the Chain Rule in Differentiation?
Answer: The chain rule in differentiation is given as a formula, which allows the calculation of derivatives of a composite function. One of its applications is found in the derivative of functions in the form of
f(x)=um
Where, u is the function of x.
Let us consider the function ‘g.’ The image by g of any of the element ‘x’ of its domain is given as g(x).
Let us consider another function ‘f.’ The image of g(x) by f is given as f[g(x)], which is also written as fog(x). The function fog is known as the composite function of the function ‘f’ and ‘g.’
2. What is Implicit Differentiation?
Answer: Implicit differentiation is applied to implicit functions. The rules that we follow are the similar rules we use when differentiating the other functions, the chain rule, the product rule, the rule for sums, and so on.
The major difference is that we can operate on an equation that we either cannot solve for the dependent variable(s), or do not want or required to solve for the dependent variable(s).
