Rules of Differentiation in Calculus

In calculus, differentiation is a method of finding the derivative of a function. The derivative of a function basically represents the slope in a graph. For example, the slope of a line 3x is 3. The slope of a constant value is zero; thus, it will be parallel to either of the axes. Some of the common functions whose derivatives are evaluated are square root, trigonometric functions, line, etc. 

For all the functions, we cannot apply the same method to find the derivatives. For example, finding the derivative of a continuous function that is continuously differentiable at every point.  Therefore, there are certain rules that are used to find these derivatives, which we will be discussing here.

What are the Rules of Differentiation?

The rules to find the derivatives of a function are:

• Power Rule
• Sum Rule
• Difference Rule
• Product Rule
• Quotient Rule
• Chain Rule

Let us discuss these rules one by one.

Power Rule

According to this rule, if the function is x is a variable that is raised to a power n, then the differentiation of x raised to the power n is given by:

(d/dx)xⁿ = nx^n-1

Let us understand by an example.

Suppose function f (x) = x⁴

Differentiating f with respect to x we get;

f’(x) = 4x^4-1 = 4x³

Sum Rule of Differentiation

As per the sum rule, the derivative of the sum of any two functions is equal to the sum of the derivative of individual functions.

If f(x)=u(x) + v(x), then;

f'(x) = u'(x) + v'(x)

We can solve an example to understand this rule.

Example: f(x) = x² + 2x

Differentiating f(x) with respect to x, we get;

f’(x) = d/dx (x² + 2x)

f’(x) = 2x + 2 = 2(x+1)

Difference Rule of Differentiation

The difference rule is the same as the sum rule for differentiation. The derivative of the differentiation of two functions will be equal to the difference of the derivative of the individual function.

If f(x)=u(x) – v(x), then;

f'(x) = u'(x) – v'(x)

Product Rule of Differentiation

If f(x) is a function that is product of any two functions u(x) and v(x), such that;

f(x) =  u(x)×v(x), 

Then the derivative of function f(x) is given by:

f′(x) = u′(x) × v(x) + u(x) × v′(x)

Example: Find the derivative of 2x (x²)

Let f(x) = 2x (x²)

Differentiating f(x), w.r.t x, we get;

f’(x) = d/dx (2x.(x²))

= d/dx(2x) (x²) + 2x d/dx (x²)

= 2.x² + 2x (2x)

= 2x² + 4x²

= 6x²

Quotient Rule of Differentiation

According to the quotient rule, if f(x) is a function, such that;

f(x) = u(x)/v(x)

Then the derivative of f(x) is given by:

f’(x) = [u’(x)v(x) – u(x)v’(x)]/v(x)²

Chain Rule of Differentiation

This rule is used for composite functions. A composite function is a function that is written inside another function. Suppose, y = f(x) = g(u) and if u = h(x), then as per the chain rule.

dy/dx = (dy/du) × (du/dx)

Example: Derivative of d/dx (1/cosx)

Let,

g(x) = cos

f(g) = 1/g(x)

Now as per chain rule;

derivative of f(g(x)) = f’(g(x))g’(x)

f’g(x) = -1/g²

g’(x) = -sin x

(1/cos(x))’ = -1/g(x)² (-sin(x)

= sin x/cos²x

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