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.
What is Differentiation in Maths?
The mathematical definition of differentiation is the change in the value of the function due to the change in the independent variable.
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:
Solved Example: Differentiate y=4x2 + x - 4 w.r.t x
Ans: Differentiating the given equation y w.r.t x.
Rules for Important Differentiation Formulas
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
Rules for Compound Functions
Some of the basic rules that need to be followed while solving and differential problem are:
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)
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)
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
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
Applications of Differentiation in Real Life Problems
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.