POWER RULE OF DERIVATIVE

Meaning of Derivative

Derivative measures amount of change in one variable with respect to the change happening in another variable.

Let us consider the two variables x and y such that y is depending on x.

Then, the derivative y with respect to x is

dy/dx

Power Rule of Derivative

Power rule or basic rule of derivative is the tool to find the derivative of y with respect to x where y is dependent variable and x is independent variable. That is, y is depending on x.

y = xⁿ

To get the derivative of xⁿ with respect to x, we have to bring the exponent n in front of x and subtract 1 from the exponent.

dy/dx = nx^{n - 1}

Example 1 :

Find dy/dx, if y = x³.

Solution :

y = x³

dy/dx = 3x^{3 - 1}

= 3x²

Example 2 :

Find dy/dx, if y = x². 

Solution :

y = x²

dy/dx = 2x^{2 - 1}

= 2x¹

= 2x

Example 3 :

Find dy/dx, if y = x. 

Solution :

y = x

y = x¹

dy/dx = 1x^{1 - 1}

= 1x⁰

= 1(1)

= 1

Example 4 :

Find dy/dx, if y = √x.

Solution :

y = √x

y = x^1/2

dy/dx = (1/2)x^{1/2 - 1}

= (1/2)x^-1/2

= 1/(2x^1/2)

= 1/(2√x)

Constant Coefficient Rule

The derivative of a variable with a constant coefficient is equal to the constant times the derivative of the variable.

That is if there is a variable x with the constant in multiplication or division, we will keep the constant as it is and find the derivative of the variable alone.

Example 5 :

Find dy/dx, if y = 5x³ + 3x².

Solution :

y = 5x³ + 3x²

Using the Power rule of derivative,

dy/dx = 5(3x^{3 - 1}) + 3(2x^{2 - 1})

= 5(3x²) + 3(2x¹)

= 15x² + 6x

Example 6 :

Find dy/dx, if y = x³/3.

Solution :

y = x³/3

Using the Power rule of derivative,

dy/dx = (3x^{3 - 1})/3

= (3x²)/3

= (3/3)x²

= x²

Example 7 :

Find dy/dx, if y = -7/x².

y = -7/x²

y = -7x ^-2

Using the Power rule of derivative,

dy/dx = -7(-2x ^{-2 - 1})

= -7(-2x ^-3)

= -7(-2/x ³)

= 14/x ³

Example 8 :

Find dy/dx, if y = 5x³ + 3.

y = 5x³ + 3

In the function above, we have two constants 5 and 3. The constant 5 is multiplied by the variable x³ and 3 is staying alone without the variable.

When we find the derivative of f(x) = 5x³ + 3, we have to keep the constant 5 as it is. Because 5 is multiplied by the variable x³. The derivative of 3 is zero, because it is not with the variable.

y = 5x³ - 3

Using the Power rule of derivative,

dy/dx = 5(3x^{3 - 1}) - 0

= 5(3x²)

= 15x²

Kindly mail your feedback to v4formath@gmail.com

We always appreciate your feedback.