Differentiation and Integration Formula

Differentiation and Integration Formulas have many important formulas. You’ll read about the formulas as well as its definition with an explanation in this article.

Definition of Differentiation

A derivative of a function related to the independent variable is called Differentiation and it is used to measure the per unit change in function in the independent variable. If y = f(x) is a function of x, then the rate of change of y per unit change in x is given as:

dy/dx

Important Differentiation Formulas

f(a) = sin a, so f'(a) = cos a
f(a) = cos a, so f'(a) = -sin a
f(a) = tan a, so f'(a) = sec2a

Definition of Integration

The integration of a function f(a) is given by F(a) and is written as

\(\int\) f(a) da = F(a) +C, where Right hand side shows integral of f(a) with respect to a

F(a) is called primitive, f(a) is called the integrand and C is constant of integration, a is variable.

Important Integration Formulas

\(\int\) sin a da = -cos a + c
\(\int\) cos a da = sin a + c
\(\int\) sec2 a da = tan a + c

Solved Examples

Example 1: Find the derivative of x2cos x.

Solution:

Let f(x) = x2 cos x

Differentiate the given function with respect to x,

f’(x) = d/dx (x2 cos x)

= x2 d/dx(cos x) + cos x d/dx(x2)

= x2 (-sin x) + cos x (2x)

= 2x cos x -x2 sin x

Example 2: Find the derivative of f(x) = x + sin x.

Solution:

Given function is f(x) = x + sin x

Differentiate with respect to x,

f’(x) = d/dx(x + sin x)

= d/dx(x) + d/dx (sin x)

= 1 + cos x

Example 3: Evaluate: ∫7 cos x dx

Solution:

∫7 cos x dx

= 7 ∫cos x dx

= 7 sin x + C

Example 4: Evaluate:  ∫6 sin 3x dx

Solution:

 ∫6 sin 3x dx

= 6  ∫ sin 3x dx …..(i)

Let u = 3x  …..(ii)

du = 3dx

dx = ⅓ du …..(iii)

From (i), (ii) and (iii),

= 6  ∫sin u (⅓) du

= (6/3)  ∫ sin u du

= 2 (-cos u) + C

Substituting u =  3x in the above step,

= -2 cos 3x + C

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