Differentiation Rules

The differentiation rules help us to evaluate the derivatives of some particular functions, instead of using the general method of differentiation. The process of differentiation or obtaining the derivative of a function has the significant property of linearity. This property makes the derivative more natural for functions constructed from the primary elementary functions, using the procedures of addition and multiplication by a constant number.

The important rules of differentiation are:

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

Let us discuss these rules one by one, with examples. Also, read the Differentiation method here at BYJU’S.

Power Rule of Derivatives

This is one of the most common rules of derivatives. If x is a variable and is raised to a power n, then the derivative of x raised to the power is represented by:

d/dx(xn) = nxn-1

Example: Find the derivative of x5

Solution: As per the power rule, we know;

d/dx(xn) = nxn-1

Hence, d/dx(x5) = 5x5-1 = 5x4

Sum Rule of Derivatives

If the function is sum or difference of two functions, then the derivative of the functions is the sum or difference of the individual functions, i.e.,

If f(x)=u(x)±v(x), then;

f'(x)=u'(x)±v'(x)

Example 1: f(x) = x + x3

Solution: By applying sum rule of derivative here, we have:

f’(x) = u’(x) + v’(x)

Now, differentiating the given function, we get;

f’(x) = d/dx(x + x3)

f’(x) = d/dx(x) + d/dx(x3)

f’(x) = 1 + 3x2

Example 2: Find the derivative of the function f(x) = 6x2 – 4x.

Solution:

Given function is: f(x) = 6x2 – 4x

This is of the form f(x) = u(x) – v(x)

So by applying the difference rule of derivatives, we get,

f’(x) = d/dx (6x2) – d/dx(4x)

= 6(2x) – 4(1)

= 12x – 4

Therefore, f’(x) = 12x – 4

Product Rule of Derivatives

According to the product rule of derivatives, if the function f(x) is the product of two functions u(x) and v(x), then the derivative of the function is given by:

If f(x) = u(x)×v(x), then:

f′(x) = u′(x) × v(x) + u(x) × v′(x)

Example: Find the derivative of x2(x+3).

Solution: As per the product rule of derivative, we know;

f′(x) = u′(x) × v(x) + u(x) × v′(x)

Here,

u(x) = x2 and v(x) = x+3

Therefore, on differentiating the given function, we get;

f’(x) = d/dx[x2(x+3)]

f’(x) = d/dx(x2)(x+3)+x2d/dx(x+3)

f’(x) = 2x(x+3)+x2(1)

f’(x) = 2x2+6x+x2

f’(x) = 3x2+6x

f’(x) = 3x(x+2)

Quotient Rule of Derivatives

If f(x) is a function, which is equal to ratio of two functions u(x) and v(x) such that;

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

Then, as per the quotient rule, the derivative of f(x) is given by;

\(\mathbf{f}^{\prime}(\mathbf{x})=\frac{\mathbf{u}^{\prime}(\mathbf{x}) \times \mathbf{v}(\mathbf{x})-\mathbf{u}(\mathbf{x}) \times \mathbf{v}^{\prime}(\mathbf{x})}{(\mathbf{v}(\mathbf{x}))^{2}}\)

Example: Differentiate f(x)=(x+2)3/√x

Solution: Given,

f(x)=(x+2)3/√x

= (x+2)(x2+4x+4)/√x

= [x3+6x2+12x+8]/x1/2

= x-1/2(x3+6x2+12x+8)

= x5/2+6x3/2+12x1/2+8x

Now, differentiating the given equation, we get;

f’(x) = 5/2x3/2 + 6(3/2x1/2)+12(1/2x-1/2)+8(−1/2x-3/2)

= 5/2x3/2 + 9x1/2 + 6x − 4x-3/2

Chain Rule of Derivatives

If a function y = f(x) = g(u) and if u = h(x), then the chain rule for differentiation is defined as;

dy/dx = (dy/du) × (du/dx)

This rule is majorly used in the method of substitution where we can perform differentiation of composite functions.

Let’s have a look at the examples given below for better understanding of the chain rule differentiation of functions.

Example 1: 

Differentiate f(x) = (x4 – 1)50

Solution:

Given,

f(x) = (x4 – 1)50

Let g(x) = x4 – 1 and n = 50

u(t) = t50

Thus, t = g(x) = x4 – 1

f(x) = u(g(x))

According to chain rule,

df/dx = (du/dt) × (dt/dx)

Here,

du/dt = d/dt (t50) = 50t49

dt/dx = d/dx g(x)

= d/dx (x4 – 1)

= 4x3

Thus, df/dx = 50t49 × (4x3)

= 50(x4 – 1)49 × (4x3)

= 200 x3(x4 – 1)49

Example 2:  

Find the derivative of f(x) = esin(2x)

Solution:

Given,

f(x) = esin(2x)

Let t = g(x) = sin 2x and u(t) = et

According to chain rule,

df/dx = (du/dt) × (dt/dx)

Here,

du/dt = d/dt (et) = et

dt/dx = d/dx g(x)

= d/dx (sin 2x)

= 2 cos 2x

Thus, df/dx = et × 2 cos 2x

= esin(2x) × 2 cos 2x

= 2 cos(2x) esin(2x)

To learn more about differentiation and differential equations, visit www.byjus.com. Also, download BYJU’S – The Learning App for interactive videos on maths concepts.

Frequently Asked Questions – FAQs

What are the basic rules of differentiation?

The basic rules of differentiation are:
Power Rule
Sum and Difference Rule
Product Rule
Quotient Rule
Chain Rule

What are the four basic derivative rules?

The four basic derivative rules include the following:
Power Rule
Sum and Difference Rule
Product Rule
Quotient Rule

What is the product rule for differentiation?

If the function f(x) is the product of two functions u(x) and v(x), then the derivative of the function is given below.
If f(x) = u(x)×v(x), then f′(x) = u′(x) × v(x) + u(x) × v′(x).
This represents the product rule for differentiation.

What is chain rule in differentiation?

If a function y = f(x) = g(u) and if u = h(x), then the chain rule for differentiation can be written as:
dy/dx = (dy/du) × (du/dx)

What is the derivative of 2x?

We know that the formula for the derivative of cx, where c is a constant. Thus, the derivative of 2x is 2.

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