CONSTANT COEFFICIENT RULE

Meaning of Derivative

Derivative is measuring changes in one variable with respect to the change in the other variable.

Let us consider two variables x and y such that y is a dependent variable and x is an independent variable.

Then, the derivative of y with respect to x is

dy/dx

Power Rule of Derivative

Power rule of derivative is the basic idea to find the derivative of y = xⁿ with respect to x.

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

dy/dx = nx^{n - 1}

Find the derivative of each of the following with respect to x.

Example 1 :

x³

Solution :

Let y = x³.

y = x³

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

= 3x²

Example 2 :

x²

Solution :

Let y = x².

y = x²

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

= 2x

Example 3 :

x

Solution :

Let y = x.

y = x

y = x¹

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

= 1x⁰

= 1(1)

= 1

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.

Find the derivative of each of the following with respect to x.

Example 4 :

5x³ + 3x²

Solution :

Let y = 5x³ + 3x².

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 5 :

x³/3

Solution :

Let y = x³/3.

y = x³/3

Using the power rule of derivative,

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

= (3x²)/3

= (3/3)x²

= x²

Example 6 :

-7/x²

Solution :

Let 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 7 :

5x³ + 3

Solution :

Let 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.