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Differentiation and Integration are two mathematical concepts that primarily form Calculus and can be understood as two concepts opposite to each other. Differentiation in calculus is defined as the instantaneous rate of change of a function with respect to one of its variables. Leibniz’s notation is commonly used for differentiation, but some other notations like Euler’s and Lagrange’s notation are also commonly used.

Leibniz’s Notation - dy/dx is defined as the infinitesimal change in y due to an infinitesimal change dx in the value of x.

Euler’s Notation - D(y) or D[f(x)], where D is the differential operator.

Lagrange’s Notation - f'(x) s also termed as prime notation.

The mathematical definition of differentiation is the change in the value of the function due to the change in the independent variable.

y=f(x)

Where y is a function of x. Any change in the value of y due to the change in the value of x is given by:

dy/dx

Solved Example: Differentiate y=4x2 + x - 4 w.r.t x

Ans: Differentiating the given equation y w.r.t x.

dy/dx=4.2.x +1-0

dy/dx=8x+1

Given below are the commonly used differentiation formulas. Please note that these are directly applicable formulas. Students are advised to remember these by heart. Let y be the function and dy/dx the derivative of the function.

y = tan x, dy/dx = sec2x

y = sin x, dy/dx = cos x

y = cos x ,dy/dx = -sin x

y = an, where a is any integer or fraction, dy/dx = nan-1

y = ex(exponential function), dy/dx=ex

y = ln(x), dy/dx = 1/x

y = k, where k is any constant, dy/dx = 0

Some of the basic rules that need to be followed while solving and differential problem are:

### Sum or Difference

If the function whose derivative is to be calculated is the sum or difference of two functions, then the derivative of the function is the sum or difference of the individual derivative of the two functions.

F(x) = f(x) ± g(x)

F’(x) =f’(x) ± g’(x)

### Product Rule

If the function in a question is the product of two individual functions, the derivative of the function is given by:

F(x) = f(x) x g(x)

F’(x) = f’(x) x g(x) + f(x) x g’(x)

### Quotient Rule

If the function whose derivative is to be calculated is the division of two functions, the derivative is calculated as follows:

F(x) = f(x)/g(x)

F’(x) = [f’(x) x g(x) - f(x) x g’(x)]/g(x)2

### Chain Rule

Sometimes to solve complex functions, a substitution method is used. If a function y = f(x) = g(z) and if z = h(x), then:

dy/dx = dy/dzdz/dx

Differentiation has real-life applications. Differentiation is defined as the change in any quantity with respect to change in other quantities. Some real-life applications are:

Acceleration is defined as the change of velocity with the change in time.

Tangent and normal to a curve is derived using a derivative function.

Maximum and minimum points of a graph to be used in the study of businesses etc.

Also used to determine any temperature variation.

FAQ (Frequently Asked Questions)

1. Differentiate between linear and non-linear functions.

Differentiation is the rate of change of the value of the function with respect to any change in the value of the independent variable. So, it is of importance to understand what type of function is in a question. Depending upon the function, the treatment of the question changes. Primarily, there are two types of functions:

**Linear Functions -**The rate of change throughout the domain of a linear function is constant. So the rate of change of function is equal to the rate of change at any instant or a given point.**Non-Linear Functions -**The rate of change varies throughout the domain of the function. So the rate of function is derived at a particular point or an instant.

2. Differentiate between integration and differentiation.

Differentiation is the instantaneous change of rate or the slope of the curve whereas integration is the area falling under the curve. Mathematically, these two algebraic expressions are opposite to each other. Given below are the differentiating points between the two concepts.

Differentiation is used to find the instantaneous changes such as velocity and also to find whether the function is increasing or decreasing. Integration gives the area under the curve.

Results are obtained through division in differentiation and summation in integration.

Both are different in the fact that they are applied differently. Also, they are the reversed processes of each other. Differentiation is the reversed process of integration and vice versa.

Differentiation can be used to find the speed of a function whereas integration is used to determine the distance travelled by a given function.

3. Explain the derivative of a function as limits with an example.

Derivative of function as limits:

If any function f(x) goes under an infinitesimal change h, then the derivative of function f(x) is given by:

f’(x) = lim_{h→}_{0}= [f(x+h)-f(x)]/h

This is also known as the derivative limit formula, where the limit of the derivative is given by,lim_{h→0}.

Solved Example:- Find the derivative of the function f=3x at x=2.

f(x)= 3x

f’(x)=lim_{h→0}[f(x+h)-f(x)]/h

f’(2)=lim_{h→0}[f(2+h)-f(2)]/h

f’(2)=lim_{h→0}[3(2+h)-3(2)]/h

f’(2)=lim_{h→0}[6+3h-6]/h

f’(2)=lim_{h→0}3h/h

f’(2)=lim_{h→0}3=3