Find the derivative of the function f(x) = x2, using the limit definition of the derivative.
Answer
Example 2
Find the derivative of the function f(x) = x3, using the limit definition of the derivative.
Answer
Example 3
Find the derivative of the function f(x) = x4, using the limit definition of the derivative.
Answer
Example 4
Now it's time for pattern-finding. We know the following functions and their derivatives:
f(x) = x2 f ' (x) = 2x
f(x) = x3 f ' (x) = 3x2
f(x) = x4 f ' (x) = 4x3
What's the pattern?
Hint
It might help to write f ' (x) = 2x as f ' (x) = 2x1.
Answer
The coefficient of the derivative is the exponent of the original function, and the exponent of the derivative is one less than the exponent of the original function.If
f(x) = x4
then
f'(x) = 4x(4 – 1) = 4x3
The derivative of the function
f(x) = xn
is
f ' (x) = nx(n – 1).
We take the exponent n from the original function. This n is the coefficient for the derivative. The value n – 1 is the exponent for the derivative.
Example 5
Find the derivative of the power function f(x) = x10.
Answer
We use the rule for finding the derivatives of power functions.
f(x) = x10
f'(x) = 10x10–1 = 10x9
Example 6
Find the derivative of the power function f(x) = x85.
Answer
f(x) = x85
f ' (x) = 85x 85 – 1 = 85x84
Example 7
Find the derivative of the power function k(x) = x3.5.
Answer
k(x) = x3.5
k ' (x) = 3.5x3.5 – 1 = 3.5x2.5
Example 8
Find the derivative of the power function k(x) = x–6.
Answer
k(x) = x–6
k'(x) = (–6)x–6–1 = –6x-7
Example 9
Find the derivative of the power function
.
Answer
This function can be rewritten as
,
therefore
.
Example 10
Find the derivative of the power function h(x) = x(π + e).
Answer
h(x) = xπ + e
The constant (π + e) is kind of unpleasant-looking, but it's still a constant.
h ' (x) = (π + e)x(π + e) – 1 = (π + e)xπ + e – 1.
Example 11
Find the derivative of the power function
.
Answer
This can be rewritten as
g(x) = x–2,
which looks a lot more like a power function.
g ' (x) = (–2)x–2 – 1 = –2x–3.
Example 12
Find the derivative of the power function
.
Answer
Again, is an ugly constant, but it is a constant nonetheless.
Example 13
What is the derivative of the power function k(x) = x0.
Answer
k(x) = x0
This is the power function. The function
k(x) = x0 = 1
is a constant function, therefore
k ' (x) = 0.
Example 14
Find the derivative of the power function
.
Answer
This function can be rewritten as
therefore
r ' (x) = ( –1)x –1 – 1 = –x–2.
Example 15
For the derivative f ' (x) = 3x2, find a possible original function.
Answer
This derivative came from a power function, the only thing we need to figure out is the exponent of the original power function.
f ' (x) = 3x2 = 3x3–1
the original function could have been
f(x) = x3.
Example 16
For the derivative f ' (x) = 8x7, find a possible original function.
Answer
f ' (x) = 8x7 = 8x 8 – 1
the original function could have been
f(x) = x8.
Example 17
For the derivative f'(x) = –3x–4, find a possible original function.
Answer
f ' (x) = –3x–4 = –3x–3–1
The original function could have been
f(x) = x–3
Example 18
For the derivative g'(x) = –9x–10, find a possible original function.
Answer
g'(x) = –9x–10 = –9x–9 – 1
the original function could have been
g(x) = x–9.
Example 19
For the derivative
,
find a possible original function, h(x).
Answer
the original function could have been
We're saying the original function "could have been" instead of "must have been" because, as we'll see later, for each derivative there are infinitely many functions with that same derivative.
.
,
.
.
.
is an ugly constant, but it is a constant nonetheless.
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