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 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 = nxn - 1
Find the derivative of each of the following with respect to x.
Example 1 :
x3
Solution :
Let y = x3.
y = x3
dy/dx = 3x3 - 1
= 3x2
Example 2 :
x2
Solution :
Let y = x2.
y = x2
dy/dx = 2x2 - 1
= 2x
Example 3 :
x
Solution :
Let y = x.
y = x
y = x1
dy/dx = 1x1 - 1
= 1x0
= 1(1)
= 1
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(3x3 - 1) + 3(2x2 - 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 = (3x3 - 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(3x3 - 1) - 0
= 5(3x2)
= 15x2
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