DERIVATIVE OF A CONSTANT

Meaning of Derivative

Derivative is the rate of change in one variable or quantity with respect to the change happening in another variable or quantity.

For example, let x be an independent variable and y be a dependent variable which is depending on x.

Then, the derivative of y with respect to x is

dy/dx

Example 1 :

An electrician charges $7 per hour for the work done by him. If y represents the cost for an electrical  work and x represents the number of hours work done, then the rate of change of y is 7/1 or 7. That is, the value of y is getting change by 7 units for every 1 unit change in x.

More clearly, the total cost of a work is increasing by $7 for every 1 hour increase in the work time.

Therefore, the derivative of y with respect to x is 7.

Example 2 :

An electrician has a fixed package of $100 per day (maximum of 10 hours). Here, if the work is done for one hour or two hours or any number of hours (within 10 hours) in a day, the electrician has to be paid $100.

Let y represent the cost of an electrical work and x represent the number of hours work done. Then the rate of change of y is 0. Because, the value of y is fixed (or constant), that is 100.

Therefore, the derivative of y with respect to x is 0.

Power Rule of Derivative

Power rule of derivative is the fundamental tool to find the derivative of a function f(x) which is in the form

f(x) = xⁿ

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

f'(x) = nx^{n - 1}

Find the derivative of each of the following :

Example 3 :

f(x) = x³

Solution :

f(x) = x³

f'(x) = 3x^{3 - 1}

= 3x²

Example 4 :

f(x) = x²

Solution :

f(x) = x²

f'(x) = 2x^{2 - 1}

= 2x¹

= 2x

Example 5 :

f(x) = 3

Solution :

f(x) = 3

f(x) = 3(1)

f(x) = 3x⁰

f'(x) = 3(0)x^{0 - 1}

= 0

Note : Since, 4 is a constant, its derivative is zero.

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 :

Example 6 :

f(x) = 2x⁵

Solution :

f(x) = 2x⁵

Using the power rule of derivative,

f'(x) = 2(5x^{5 - 1})

= 10x⁴

Example 7 :

f(x) = 5x³ + 3x²

Solution :

f(x) = 5x³ + 3x²

Using the power rule of derivative,

f'(x) = 5(3x^{3 - 1}) + 3(2x^{2 - 1})

= 5(3x²) + 3(2x¹)

= 15x² + 6x

Example 8 :

f(x) = x³/3

Solution :

f(x) = x³/3

Using the power rule of derivative,

f'(x) = (3x^{3 - 1})/3

= (3x²)/3

= (3/3)x²

= x²

Example 9 :

f(x) = -7/x²

Solution :

f(x) = -7/x²

f(x) = -7x ^-2

Using the power rule of derivative,

f'(x) = -7(-2x ^{-2 - 1})

= -7(-2x ^-3)

= -7(-2/x ³)

= 14/x ³

Example 10 :

f(x) = 5x³ + 3

Solution :

In the given function, 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.

f(x) = 5x³ - 3

Using the power rule of derivative,

f'(x) = 5(3x^{3 - 1}) - 0

= 5(3x²)

= 15x²

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