## Differentiation

A **Differentiation formulas** list has been provided here for students so that they can refer to these to solve problems based on differential equations. This is one of the most important topics in higher class Mathematics. The general representation of the derivative is **d/dx**.

This formula list includes derivatives for constant, trigonometric functions, polynomials, hyperbolic, logarithmic functions, exponential, inverse trigonometric functions etc. Based on these, there are a number of examples and problems present in the syllabus of Class 11 and 12, for which students can easily write answers.

## Differentiation Formulas List

### Differentiation Formulas for Trigonometric Functions

Trigonometry is the concept of relation between angles and sides of triangles. Here, we have 6 main ratios, such as, sine, cosine, tangent, cotangent, secant and cosecant. You must have learned about basic trigonometric formulas based on these ratios. Now let us see the formulas for **derivatives of trigonometric functions** and hyperbolic functions.

### Differentiation Formulas for Inverse Trigonometric Functions

Inverse trigonometry functions are the inverse of trigonometric ratios. Let us see the formulas for **derivatives of inverse trigonometric functions**.

### Other Differentiation Formulas

### Definition of Derivatives

The geometrical meaning of the derivative of y = f(x) is the slope of the tangent to the curve y = f(x) at ( x, f(x)). The first principle of differentiation is to compute the derivative of the function using the limits. Let a function of a curve be y = f(x). Let us take a point P with coordinates(x, f(x)) on a curve. Take another point Q with coordinates (x+h, f(x+h)) on the curve. Now PQ is the secant to the curve. The slope of a curve at a point is the slope of the tangent line at that point. We know, slope of the secant line is

We want h to be as small as possible to get the slope of the tangent. We have y = f(x). There is an incremental change in x, denoted as Δx. Then there exists an incremental change in y, denoted as Δy.

Then y + Δy = f(x + Δx)

f(x) + Δy = f(x + Δx)

Δy = f(x + Δx) – f(x)

Dividing by Δx on both the sides,

This derivative of f(x) at a quantifies the change in f(x) with respect to x. This process of computing the derivative of a function is called differentiation.

Thus **definition of derivative** is as follows: If f is a real-valued function of a real variable defined on an open interval I and if y = f(x) is a differentiable function of x, then

### Differentiation of Elementary Functions

- The derivative of a constant function is 0. if y = k, where k is a constant, then y’ = 0
- The derivative of a power function: If y = x
^{n }, n > 0. Then y’ = n x^{n-1} - The derivative of logarthmic functions: If y = lnee x, then y’ = 1/x and if y = logaa x, then y’ = 1/[(log a) x]
- The derivative of an exponential function: If y = a
^{ x }, y = a^{x }log a

### Implicit Differentiation

Let f(x,y) be a function in the form of x and y. If we cannot solve for y directly, we use implicit differentiation. Suppose f(x,y) =0, then differentiate this function with respect to x and collect the terms containing dy/dx at one side and then find dy/dx.

For example, let us find dy/dx if x^{2} +y^{2} =1.

We differentiate both sides of the equation.

d/dx. x^{2} + d/dx. y^{2 }= d/dx.1

2x + 2y.dy/dx = 0

dy/ dx = -x/y

### Logarithmic Differentiation Functions

If a function is the product and quotient of functions, as in y = f1(x).f2(x)…..g1(x).g2(x)….f1(x).f2(x)…..g1(x).g2(x)…. we first take the logarithm and then differentiate it. If a function is in the form of an exponent of a function over another, as in [f(x)] ^{g(x) }then we take the logarithm of the function f(x) (to base e) and then differentiate it.

For example, if y = x^{x }, then log y = x log x

1/y. dy/dx = log x + 1

dy/dx = y. (logx + 1)

= x^{x }(logx + 1)

## Higher-Order Differentiation

We find higher-order derivatives on successive differentiation. The

^{n}(x) is used in the power series. For example, the rate of change of displacement is the velocity. The second derivative of displacement is the acceleration and the third derivative is called the jerk.

Consider a function y = f(x) = x^{5 }– 3x^{4 }+ x

f^{1}(x) = 5x^{4 }– 12x^{3 }+ 1

f^{2}(x) = 20x^{3 }– 36 x^{2}

f^{3}(x) = 60x^{2 }– 72 x

f^{4}(x) = 120x -72

### Partial Differentiation

The partial differential coefficient of f(x,y) with respect to x is the ordinary differential coefficient of f(x,y) when y is regarded as a constant. It is written as 𝛿y/ 𝛿x. For example, if z = f(x,y) = x^{4} + y^{4}+3xy^{2} +x^{2}y +x + 2y, then we consider y as constant to find 𝛿f/ 𝛿x and consider x as constant to find 𝛿f/ 𝛿y. Thus we find the partial derivatives of the function.

𝛿f/ 𝛿x = 4x^{3} +3y^{2} +2xy +1

𝛿f/ 𝛿y = 4y^{3} + 6xy + x^{2} + 2

If f(x,y) is a function of two variables such that 𝛿f/ 𝛿x and 𝛿f/ 𝛿y both exist. Then we have the partial derivatives as follows.

𝛿f/ 𝛿x wrt x = 𝛿^{2 }f/ 𝛿x^{2 }or fxxfxx

𝛿f/ 𝛿y wrt y= 𝛿^{2 }f/ 𝛿y^{2 }or fyyfyy

𝛿f/ 𝛿x wrt x = 𝛿^{2 }f/ 𝛿x𝛿y ^{ }or fxyfxy

𝛿f/ 𝛿y wrt y= 𝛿^{2 }f/ 𝛿x𝛿y ^{ }or fyx

**Important Notes**

- Differentiation of a function is finding the rate of change of the function with respect to another quantity.

in x.

The process of finding the derivatives of the function, if the limit exists, is called differentiation. The derivative of a function is given as dy/dx or y’ or f'(x).

Differentiability implies continuity, but its converse is not true.

### What is Differentiation in Maths

In Mathematics, Differentiation can be defined as a derivative of a function with respect to an independent variable. Differentiation, in calculus, can be applied to measure the function per unit change in the independent variable.

Let y = f(x) be a function of x. Then, the rate of change of “y” per unit change in “x” is given by:

**dy / dx**

If the function f(x) undergoes an infinitesimal change of ‘h’ near to any point ‘x’, then the derivative of the function is defined as

### Derivative of Function As Limits

If we are given with real valued function (f) and x is a point in its domain of definition, then the derivative of function, f, is given by:

f'(a) = lim_{h→0}[f(x+h)-f(x)]/h

provided this limit exists.

Let us see an example here for better understanding.

**Example: Find the derivative of f=2x, at x =3.**

Solution: By using the above formulas, we can find,

f'(3) = lim_{h→0}[f(3+h)-f(3]/h = lim_{h→0}[2(3+h)-2(3)]/h

f'(3) = lim_{h→0}[6+2h-6]/h

f'(3) = lim_{h→0}2h/h

f'(3) = lim_{h→0}2 = 2

### Notations

When a function is denoted as y=f(x), the derivative is indicated by the following notations.

**D(y) or D[f(x)]**is called Euler’s notation.**dy/dx**is called Leibniz’s notation.**F’(x)**is called Lagrange’s notation.

The meaning of differentiation is the process of determining the derivative of a function at any point.

### Linear and Non-Linear Functions

Functions are generally classified in two categories under Calculus, namely:

**(i) Linear functions**

**(ii) Non-linear functions**

A linear function varies with a constant rate through its domain. Therefore, the overall rate of change of the function is the same as the rate of change of a function at any point.

However, the rate of change of function varies from point to point in case of non-linear functions. The nature of variation is based on the nature of the function.

The rate of change of a function at a particular point is defined as a **derivative** of that particular function.

## Differentiation Formulas

The important Differentiation formulas are given below in the table. Here, let us consider f(x) is a function and f'(x) is the derivative of the function.

## Differentiation Rules

The basic differentiation rules that need to be followed are as follows:

- Sum and Difference Rule
- Product Rule
- Quotient Rule
- Chain Rule

### Sum or Difference Rule

If the function is sum or difference of two functions, the derivative of the functions is the sum or difference of the individual functions, i.e.,

**If f(x) = u(x) ± v(x)**

### Product Rule

As per the product rule, if the function f(x) is product of two functions u(x) and v(x), the derivative of the function is,

**If f(x)=u(x)×v(x)f(x)=u(x)×v(x)**

**then, f′(x)=u′(x)×v(x)+u(x)×v′(x)f′(x)=u′(x)×v(x)+u(x)×v′(x)**

### Quotient rule

If the function f(x) is in the form of two functions [u(x)]/[v(x)], the derivative of the function is

### Chain Rule

If a function y = f(x) = g(u) and if u = h(x), then the chain rule for differentiation is defined as,

This plays a major role in the method of substitution that helps to perform differentiation of composite functions.

### Real-Life Applications of Differentiation

With the help of differentiation, we are able to find the rate of change of one quantity with respect to another. Some of the examples are:

- Acceleration: Rate of change of velocity with respect to time
- To calculate the highest and lowest point of the curve in a graph or to know its turning point, the derivative function is used
- To find tangent and normal to a curve

### Solved Examples

**Q.1: Differentiate f(x) = 6x ^{3}-9x+4 with respect to x.**

Solution: Given: f(x) = 6x

^{3}-9x+4

On differentiating both the sides w.r.t x, we get;

f'(x) = (3)(6)x^{2} – 9

f'(x) = 18x^{2} – 9

This is the final answer.

**Q.2: Differentiate y = x(3x ^{2} – 9)**

Solution: Given, y = x(3x^{2} – 9)

y = 3x^{3} – 9x

On differentiating both the sides we get,

dy/dx = 9x^{2} – 9

This is the final answer.

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### Examples of Differentiation

**Example 1. Find the differentiation of y = x ^{3 }+ 5 x^{2}+ 3x + 7.**

**Solution:**

Given y = x^{3 }+ 5 x^{2}+ 3x + 7

We differentiate y with respect to x.

Using the differentiation formula of power rule, we get dy/dx = dy/dx( x^{3 }+ 5 x^{2}+ 3x + 7)

= d(x^{3})/dx + d(5 x^{2 })/dx + d(3x)/dx + d(7)/dx

dy/dx = 3 x^{2 }+ 5(2x) + 3 dy/dx + 0

= 3 x^{2 }+ 10 x + 3

**Answer: dy/dx = 3 x ^{2 }+ 10 x + 3**

**Example 2. Find the differentiation of y = cos(tan x)**

**Solution:**

Given: y = cos(tan x)

We differentiate y with respect to x.

Let u = tan x

y = cos u

Using the chain rule of differentiation,

dy/dx = dy/du . du/dx

dy/dx = d(cos u)/du . d(tan x)/dx

= -sin u . sec^{2} x

= -sin (tan x) . sec^{2} x

**Answer: dy/dx = -sin (tan x) . sec ^{2} x**

## Frequently Asked Questions – FAQs

### What are the formulas of differentiation?

The formulas of differentiation that helps in solving various differential equations include:

Derivatives of basic functions

Derivatives of Logarithmic and Exponential functions

Derivatives of Trigonometric functions

Derivatives of Inverse trigonometric functions

Differentiation rules

### What are the basic rules of differentiation?

The basic rule of differentiation are:

Power Rule: (d/dx) (x^n ) = nx^{n-1}

Sum Rule: (d/dx) (f ± g) = f’ ± g’

Product Rule: (d/dx) (fg)= fg’ + gf’

Quotient Rule: (d/dx) (f/g) = [(gf’ – fg’)/g^2]

### What are the derivatives of trigonometric functions?

The derivatives of six trigonometric functions are:

(d/dx) sin x = cos x

(d/dx) cos x = -sin x

(d/dx) tan x = sec^2 x

(d/dx) cosec x = -cosec x cot x

(d/dx) sec x = sec x tan x

(d/dx) cot x = -cosec^2 x

### What is d/dx?

The general representation of the derivative is d/dx. This denotes the differentiation with respect to the variable x.

### What is a UV formula?

(d/dx)(uv) = v(du/dx) + u(dv/dx)

This formula is used to find the derivative of the product of two functions.

### What is Differentiation?

The instantaneous rate of change of a function with respect to another quantity is called differentiation. For example, speed is the rate of change of displacement at a certain time. If y = f(x) is a differentiable function of x, then dy/dx = f'(x) =

### How Do You Perform Differentiation in Math?

Differentiation is done by applying the techniques of known differentiation formulas and differentiation rules in finding the derivative of a given function.

### What Are The Basics of Differentiation?

The process of finding the derivative of a function is called differentiation. The three basic derivatives are differentiating the algebraic functions, the trigonometric functions, and the exponential functions.

### Give an Example of Differentiation.

The rate of change of displacement with respect to time is the velocity. This is an example of differentiation. Velocity is the first derivative of displacement. Acceleration is the second derivative of displacement.

### What Are Differentiation Formulas?

The differentiation formula is used to find the derivative or rate of change of a function. if y = f(x), then the derivative dy/dx = f'(x) =

### How Do You Use Differentiation Formula?

The derivative of a function is found by applying limits to the function as per the first principle of differentiation. The derivative

### What Are The Differentiation Rules?

There are different rules followed in differentiating a function. The differentiation rules are power rule, chain rule, quotient rule, and the constant rule.

- Sum Rule: If y = u(x) ± v(x), then dy/dx = du/dx ± dv/dx.
- Product Rule: If y = u(x) × v(x), then dy/dx = u.dv/dx + v.du/dx
- Quotient Rule: If y = u(x) ÷ v(x), then dy/dx = (v.du/dx- u.dv/dx)/ v
^{2} - Chain Rule: Let y = f(u) be a function of u and if u=g(x) so that y = f(g(x), then d/dx(f(g(x))= f'(g(x))g'(x)
- Constant Rule: y = k f(x), k ≠ 0, then d/dx(k(f(x)) = k d/dx f(x).

### What Are The Applications of Differentiation Formulas?

We use the differentiation formulas to find the maximum or minimum values of a function, the velocity and acceleration of moving objects, and the tangent of a curve.

### What is The Differentiation of a Constant?

The differentiation of a constant is 0 as per the constant rule of differentiation.

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