Differentiation - Mathematical Tools Physics Class 11 (Concepts and Questions)

In this article, we will learn about exponential and logarithmic functions. After that, we shall explore the basic concepts of differential calculus used as mathematical tools to prepare for NEET physics syllabus.

Check the latest NEET Syllabus for the 2023/2024 exams.

Table of Contents

Exponential Functions
- Natural Exponential Functions
- Rules for Exponential Functions
Logarithmic Functions
Differentiation
- Rules
- Chain Rule

Exponential Functions

A function is simply a rule that relates two or more variables. Depending on the type of rule, we have different types of functions. If the rule is a polynomial, then the function is called an algebraic function. For example, linear functions, quadratic functions, cubic functions, etc. If the rule has trigonometric ratios, the function is called trigonometric function.

Also check out Trigonometry – Mathematical Tools for Physics Class 11 (Concepts and Questions)

We define the exponential function as a real-valued function. f: R → (0, ∞) is called an exponential function, if for a > 0 we have f(x) = a^x. In simple words, the rule for the function must be an exponent.

An application of exponential function in physics can be radioactivity growth or decay. It can also be used to describe population growth.

Natural Exponential Function

A real-valued function of type f(x) = e^x, where e is known as the Euler’s constant (e ≈ 2.718), is called the natural exponential function.

The figure below shows the graph of an exponential function.

Graph of exponential function

From the above graph, we notice that for smaller values of x, the slope of the graph changes slowly, whereas, for the bigger values, it changes drastically.

Rules for Exponential Functions

There are certain rules regarding the bases and indices of an exponential function that we must follow.

Product Rule: a^m × aⁿ = a^{m + n}; the base is the same, but the powers are different.
Quotient Rule: a^m ÷ aⁿ = a^{m – n}; the base is the same, but the powers are different.
Power Rule: (a^m)ⁿ = a^{m × n}
Zero Exponent: Any number raised to the power zero is equal to 1, that is, a⁰ = 1.
Negative Exponent: a^–1 = 1/a.
Power of a Product: (a × b)^m = a^m × b^m
Power of Quotient: (a/b)^m = a^m/b^m

Logarithmic Functions

If x and a are two positive real numbers such that a ≠ 1, then y = log_a x is known as the logarithmic function with base a. Thus,

y = f(x) = log_a x ⇒ a^y = x

Where y is called the logarithm, a is the base, and x is the argument. To understand the above meaning of the above function, let us take an example,

Equation Solution

2^x = 2 x = 1

2^x = 4 x = 2

2^x = 32 x = 5

2^x = 40 x = ?

2^x = 6 x = 6

In the above example, we see that it is a bit difficult to find the value of x in case 2^x = 40, but using logarithm, we easily find the value of x.

That is, 2^x = 40 ⇒ log₂ 40 = x.

If the logarithm has a base 10, it is called standard logarithms whereas if the base is e, it is known as natural logarithms.

Properties of Logarithmic Functions

Following are the properties of logarithmic functions which are directly implied from the properties of exponents.

Product Rule: log_a mn = log_a m + log_a n
Quotient Rule: log_a (m/n) = log_a m – log_a n
Power Rule: log_a m^p = p log_a m

Note: log_a 1 = 0 as, a^x = 1 ⇒ x = 1.

Differentiation

Let y be the dependent variable which is a function of x (independent variable). Differentiation is the rate of change of y with respect to x.

Let Δx be the small change in x due to which there is a change of Δy in y. Then,

the rate of change in y with respect to x = Δy/Δx

As Δ → 0, Δy/Δx → dy/dx is known as the derivative of y with respect to x.

Thus, the instantaneous rate of change is given by

\(\begin{array}{l}\displaystyle \lim_{\Delta x \to 0}\frac{\Delta y}{\Delta x}=\frac{dy}{dx}\end{array} \)

Rules for Differentiation of Functions

For algebraic function xⁿ,

\(\begin{array}{l}\frac{d}{dx}x^{n}=nx^{(n-1)}\end{array} \)

Let us take an example, 2x⁵

\(\begin{array}{l}\frac{d}{dx}2x^{5}=2\times 5x^{5-1}=10x^{4}\end{array} \)

For trigonometric functions

\(\begin{array}{l}1.\:\:\frac{d}{dx}\;sin\;x= cos\;x\end{array} \)

\(\begin{array}{l}2.\:\:\frac{d}{dx}\;cos\;x= -sin\;x\end{array} \)

\(\begin{array}{l}3.\:\:\frac{d}{dx}\;tan\;x= -sec\;x\end{array} \)

\(\begin{array}{l}4.\:\:\frac{d}{dx}\;cosec\;x= -cosec\;x\;cot\;x\end{array} \)

\(\begin{array}{l}5.\:\:\frac{d}{dx}\;sec\;x= sec\;x\;tan\;x\end{array} \)

\(\begin{array}{l}6.\:\:\frac{d}{dx}\;cot\;x= -cosec^{2}\;x\end{array} \)

For logarithmic functions

\(\begin{array}{l}\frac{d}{dx}\;log_{e}\;x= \frac{1}{x}\end{array} \)

For exponential function

\(\begin{array}{l}1.\:\: \frac{d}{dx}\;e^{x}= e^{x}\end{array} \)

\(\begin{array}{l}2.\:\: \frac{d}{dx}\;a^{x}= a^{x}log_{e}a\end{array} \)

Chain Rule for Differentiation

Let g be a function of x, that is u = g(x) and f be a function of g, then

\(\begin{array}{l}\frac{d}{dx}f(u)=\frac{d}{du}f(u)\frac{du}{dx}\end{array} \)

For example, we have to differentiate the function log_e(sin x).

Let sin x = u then log_e u

\(\begin{array}{l}\frac{d}{dx}log_{e}u=\frac{d}{du}log_{e}u\frac{du}{dx}\end{array} \)

\(\begin{array}{l}= \frac{1}{sin\;x}.\frac{d}{dx}sin\;x=\frac{cos\;x}{sin\;x}=cot\;x\end{array} \)

Recommended Video

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