Derivative of a Function with another Function

For the process of differentiation of a function with another function, there is a particular rule called the chain rule. This article exemplifies the rule. It’s critical to do a lot of practice activities to learn the concepts described in the article so that they can become second nature. You will be able to: explain what a function of a function means after reading this text,  spell out the chain rule  and  use differentiation to find the derivative of a function with another a function

The chain rule ( to find derivative of a function with another function )

The chain rule is a formula in calculus that computes the derivative of a combination of two or more functions. In other words, if f and g are both functions, the chain rule explains the derivatives of the composite function f g in respect to the derivative of f and g.

The chain rule has the more memorable “symbolic cancellation” form in German mathematician, Gottfried Wilhelm Leibniz’s notation, which employs d/dx instead of D and so permits differentiation concerning distinct variables to be made explicit:d(f(g(x))/dx = df/dg dg/dx.

The chain rule was already known since the end of the 1700s when Isaac Newton and sir Gottfried Leibniz first discovered calculus. Calculations involving computing the derivatives of complicated expressions, such as those encountered in many physics applications, are made easier with this method.

The formula of the chain rule

The Chain Rule: dy/dx = dy /du  du/ dx

Finding derivative of a function with another function

Take a look at the equation cos x 2. We can see right away that this is not the same as the simple cosine function: cos x. We’re looking for the cosine of x 2 rather than the cosine of x. A ‘function of a function is what we term such an expression. Assume we possess 2 functions, f(x) and g(x), in general (x). T hen y = f(g(x)) is also a function of a function. The function f is indeed the sine function, and the function g seems to be the square function in our case.

We might more mathematically identify them by stating : f(x) = cos x g(x) = x 2, resulting in f(g(x)) = f(x 2) = cos x 2. Let’s take a closer look at another scenario. Assume that f is the square function while g is the sine function this time. To put it another way, if f(x) = x 2 and g(x) = cos x, then f(g(x)) = f(cos x) = (cos x) 2 (cos x) 2 is frequently written as cos2 x. As a result, cos2 x is now a function of a function. We’ll learn how to distinguish such a function in the next section.

Examples 

Question 1: 

Differentiate the following:

 f(x) = (x4 – 1)50

Solution:

Given the fact,

f(x)=(x4 – 1)50

Let g(x)=x –1 and n = 50

u(t)=t50

Therefore, t=g(x)=x4 – 1

f(x)=u(g(x))

According to the chain rule,

df/dx=(du/dt) × (dt/dx)

Here,

du/dt=d/dt (t50)=50t49

dt/dx=d/dx g(x)

= d/dx (x4 – 1)

= 4×3

Thus, df/dx=50t49 × (4×3)

= 50(x4 – 1)49 ×(4×3)

= 200 x3(x4 – 1)49

Example 2:  

Find the derivative of the following:

f(x)=esin(2x)

Solution:

Given,

f(x)=esin(2x)

Let t=g(x)=sin 2x and u(t)=et

According to the chain rule,

df/dx=(du/dt) × (dt/dx)

Here,

du/dt=d/dt (et) = et

dt/dx=d/dx g(x)

= d/dx (sin 2x)

= 2 cos 2x

Therefore, df/dx=et × 2 cos 2x

= esin(2x) × 2 cos 2x

= 2 cos(2x) esin(2x)

Conclusion

To summarise today’s article, The chain rule is a calculus formula for calculating the derivatives of a pair of 2 or more functions. In Gottfried Wilhelm Leibniz’s writing, the chain rule takes the memorable “symbolic cancellation” version, which uses d/dg dg/dx. In short, the chain rule is used to calculate and find the derivative of a function with another function. As is the case with any mathematical concept, practice is really important to win the game in any exam. Take a look at the above examples and practise some on your own.