Power rule
At this point, we understand the idea of the derivative, and w
to find it using the definition.
While the definition of the derivative can always be used to f
derivative of a function, it’s not usually the most efficient way
the derivative.
It’ll be faster for us to use the derivative rules we’re about to
lesson, we’ll look at the first of those derivative rules, which is
rule.
The power rule
The power rule lets us take the derivative of power functions
functions are things like x 2, 3x 4, 6x 5, etc.
Power rule tells us that, to take the derivative of a function lik
we just multiply the exponent by the coefficient, and then su
the exponent.
Formally, power rule says that, for any function of the form a
derivative will be
d
(a x n) = (a ⋅ n)x n−1
,For instance, to find the derivative of 3x 4, we’ll bring down th
4 to multiply it by the coefficient of 3, and we’ll subtract 1 from
exponent of 4. So the derivative would be
3(4)x 4−1
12x 3
We can also use power rule to find the derivative of polynom
combinations of power functions.
Example
Find the derivative of the function.
f (x) = 7x 3 + 2x 2 − 3x
We can use power rule to take the derivative of the function
time. We’ll apply the power rule to each term.
f′(x) = 7(3)x 3−1 + 2(2)x 2−1 − 3(1)x 1−1
f′(x) = 21x 3−1 + 4x 2−1 − 3x 1−1
f′(x) = 21x 2 + 4x 1 − 3x 0
f′(x) = 21x 2 + 4x − 3(1)
, We want to notice a couple of things about this last example
applying power rule, we ended up with 4x 1 for the second ter
derivative. It’s not necessary to write an exponent when the e
it’s implied. So 4x 1 can be written more simply as just 4x.
Second, we used power rule to take the derivative of the thir
To apply power rule, we had to realize that −3x is equivalent
that we could use the exponent of 1. After applying power ru
normal, to −3x 1, we got −3x 0. Anything raised to the 0 power
so x 0 turns into 1.
The takeaway here is that the derivative of any term where t
is 1, will be equal to the coefficient. So the derivative of −3x is
derivative of 7x is 7, and the derivative of x is 1.
The derivative of a constant
Similarly, power rule tells us that the derivative of any consta
be 0. In other words, the derivative of −3 is 0, the derivative o
the derivative of 1 is 0.
As an example, take the constant −7. We can rewrite −7 as −
equivalent to 1, and multiplying 1 doesn’t change the value o
If we then use the power rule to take the derivative of the con
At this point, we understand the idea of the derivative, and w
to find it using the definition.
While the definition of the derivative can always be used to f
derivative of a function, it’s not usually the most efficient way
the derivative.
It’ll be faster for us to use the derivative rules we’re about to
lesson, we’ll look at the first of those derivative rules, which is
rule.
The power rule
The power rule lets us take the derivative of power functions
functions are things like x 2, 3x 4, 6x 5, etc.
Power rule tells us that, to take the derivative of a function lik
we just multiply the exponent by the coefficient, and then su
the exponent.
Formally, power rule says that, for any function of the form a
derivative will be
d
(a x n) = (a ⋅ n)x n−1
,For instance, to find the derivative of 3x 4, we’ll bring down th
4 to multiply it by the coefficient of 3, and we’ll subtract 1 from
exponent of 4. So the derivative would be
3(4)x 4−1
12x 3
We can also use power rule to find the derivative of polynom
combinations of power functions.
Example
Find the derivative of the function.
f (x) = 7x 3 + 2x 2 − 3x
We can use power rule to take the derivative of the function
time. We’ll apply the power rule to each term.
f′(x) = 7(3)x 3−1 + 2(2)x 2−1 − 3(1)x 1−1
f′(x) = 21x 3−1 + 4x 2−1 − 3x 1−1
f′(x) = 21x 2 + 4x 1 − 3x 0
f′(x) = 21x 2 + 4x − 3(1)
, We want to notice a couple of things about this last example
applying power rule, we ended up with 4x 1 for the second ter
derivative. It’s not necessary to write an exponent when the e
it’s implied. So 4x 1 can be written more simply as just 4x.
Second, we used power rule to take the derivative of the thir
To apply power rule, we had to realize that −3x is equivalent
that we could use the exponent of 1. After applying power ru
normal, to −3x 1, we got −3x 0. Anything raised to the 0 power
so x 0 turns into 1.
The takeaway here is that the derivative of any term where t
is 1, will be equal to the coefficient. So the derivative of −3x is
derivative of 7x is 7, and the derivative of x is 1.
The derivative of a constant
Similarly, power rule tells us that the derivative of any consta
be 0. In other words, the derivative of −3 is 0, the derivative o
the derivative of 1 is 0.
As an example, take the constant −7. We can rewrite −7 as −
equivalent to 1, and multiplying 1 doesn’t change the value o
If we then use the power rule to take the derivative of the con
