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
F(a) is called primitive, f(a) is called the integrand and C is constant of integration, a is variable.
Important Integration Formulas
\(\begin{array}{l}\int\end{array} \) sin a da = -cos a + c |
\(\begin{array}{l}\int\end{array} \) cos a da = sin a + c |
\(\begin{array}{l}\int\end{array} \) sec2 a da = tan a + c |
Solved Examples
Example 1: Find the derivative of x2cos x.
Solution:
Let f(x) = x2Â cos x
Differentiate the given function with respect to x,
f’(x) = d/dx (x2 cos x)
= x2Â d/dx(cos x) + cos x d/dx(x2)
= x2Â (-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 =Â 3x in the above step,
= -2 cos 3x + C
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