Maharashtra Board Class 12 Maths Part 2 Chapter 1 Differentiation PDF Download

1. Differentiation

Let us Study

Derivatives of Composite functions.

Geometrical meaning of Derivative.

Derivatives of Inverse functions.

Logarithmic Differentiation.

Derivatives of Implicit functions.

Derivatives of Parametric functions.

Higher order Derivatives.

Let us Recall

The derivative of f(x) with respect to x, at x = a is given by \(f'(a) = \lim_{h \to 0} \frac{f(a+h) - f(a)}{h}\)

The derivative can also be defined for f(x) at any point x on the open interval as \(f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}\). If the function is given as y = f(x) then its derivative is written as \(\frac{dy}{dx} = f'(x)\).

For a differentiable function y = f(x) if δx is a small increment in x and the corresponding increment in y is δy then \(\lim_{\delta x \to 0} \frac{\delta y}{\delta x} = \frac{dy}{dx}\).

Derivatives of some standard functions.

Rules of Differentiation

If u and v are differentiable functions of x such that

(i) y = u ± v then \(\frac{dy}{dx} = \frac{du}{dx} \pm \frac{dv}{dx}\)

(ii) y = uv then \(\frac{dy}{dx} = u\frac{dv}{dx} + v\frac{du}{dx}\)

(iii) y = \(\frac{u}{v}\) where v ≠ 0 then \(\frac{dy}{dx} = \frac{v\frac{du}{dx} - u\frac{dv}{dx}}{v^2}\)

Introduction

The history of mathematics presents the development of calculus as being accredited to Sir Isaac Newton (1642-1727) an English physicist and mathematician and Gottfried Wilhelm Leibnitz (1646-1716) a German physicist and mathematician. The Derivative is one of the fundamental ideas of calculus. It is all about rate of change in a function. We try to find interpretations of these changes in a mathematical way. The symbol δ will be used to represent the change, for example δx represents a small change in the variable x and it is read as "change in x" or "increment in x". δy is the corresponding change in y if y is a function of x.

We have already studied the basic concept, derivatives of standard functions and rules of differentiation in previous standard. This year, in this chapter we are going to study the geometrical meaning of derivative, derivatives of Composite, Inverse, Logarithmic, Implicit and Parametric functions and also higher order derivatives. We also add some more rules of differentiation.

Let us Learn

1.1.1 Derivatives of Composite Functions (Function of another function)

So far we have studied the derivatives of simple functions like sin x, log x, \(e^x\) etc. But how about the derivatives of \(\sin \sqrt{x}\), log(sin(x² + 5)) or \(e^{\tan x}\) etc? These are known as composite functions. In this section let us study how to differentiate composite functions.

1.1.2 Theorem

If y = f(u) is a differentiable function of u and u = g(x) is a differentiable function of x such that the composite function y = f[g(x)] is a differentiable function of x then \(\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}\).

Proof

Given that y = f(u) and u = g(x). We assume that u is not a constant function. Let there be a small increment in the value of x say δx then δu and δy are the corresponding increments in u and y respectively.

As δx, δu, δy are small increments in x, u and y respectively such that δx ≠ 0, δu ≠ 0 and δy ≠ 0.

We have \(\frac{\delta y}{\delta x} = \frac{\delta y}{\delta u} \times \frac{\delta u}{\delta x}\).

Taking the limit as δx → 0 on both sides we get,

\(\lim_{\delta x \to 0} \frac{\delta y}{\delta x} = \lim_{\delta x \to 0} \frac{\delta y}{\delta u} \times \lim_{\delta x \to 0} \frac{\delta u}{\delta x}\) . . . . . (I)

Since y is a differentiable function of u and u is a differentiable function of x, we have,

\(\lim_{\delta u \to 0} \frac{\delta y}{\delta u} = \frac{dy}{du}\) and \(\lim_{\delta x \to 0} \frac{\delta u}{\delta x} = \frac{du}{dx}\) . . . . . (II)

From (I) and (II), we get

\(\lim_{\delta x \to 0} \frac{\delta y}{\delta x} = \frac{dy}{du} \times \frac{du}{dx}\) . . . . . (III)

The R.H.S. of (III) exists and is finite implies L.H.S. of (III) also exists and is finite. Then equation (III) becomes,

\(\frac{dy}{dx} = \frac{dy}{du} \times \frac{du}{dx}\)

Note

1. The derivative of a composite function can also be expressed as follows. y = f(u) is a differentiable function of u and u = g(x) is a differentiable function of x such that the composite function y = f[g(x)] is defined then \(\frac{dy}{dx} = f'[g(x)] \cdot g'(x)\).

2. If y = f(v) is a differentiable function of v and v = g(u) is a differentiable function of u and u = h(x) is a differentiable function of x then \(\frac{dy}{dx} = \frac{dy}{dv} \times \frac{dv}{du} \times \frac{du}{dx}\).

3. If y is a differentiable function of \(u_1\), \(u_i\) is a differentiable function of \(u_{i+1}\) for i = 1, 2, ..., n-1 and \(u_n\) is a differentiable function of x, then \(\frac{dy}{dx} = \frac{dy}{du_1} \times \frac{du_1}{du_2} \times \frac{du_2}{du_3} \times ... \times \frac{du_{n-1}}{du_n} \times \frac{du_n}{dx}\).

This rule is also known as Chain rule.

Teacher's Note

The chain rule helps us find derivatives of nested functions. In real life, like a train connected to many bogies, composite functions are functions connected together.

Exam Trick

Remember: In chain rule, multiply the derivatives like a chain. First derivative of outer function times the derivative of inner function. Always work from outside in.

Points to Remember

Composite function has one function inside another function.
Chain rule is used to differentiate composite functions.
Always identify the outer and inner functions first.
Multiply derivatives like links in a chain.
This rule works for any number of nested functions.

This is a preview of the first 3 pages. To get the complete book, click below.