CK-12-Calculus

http://www.ck12.org Chapter 2. Derivatives

Note that this proof depends on using the binomial theorem from precalculus.
Example 2:
Iff(x) =x^3 , then

f′(x) = 3 x^3 −^1 = 3 x^2

and

d

dx[x] =^1 ·x

1 − (^1) =x (^0) = 1 ,
d
dx[
√x] = d
dx[x
1 / (^2) ] =^1
2 x
1 / 2 − (^1) =^1
2 x
− 1 / (^2) =^1
2 x^1 /^2 =

1

2 √x,

d

dx

[ 1

x^3

]

=dxd[x−^3 ] =− 3 x−^3 −^1 =− 3 x−^4 =−x 43 ,

The Power Rule and a Constant
Ifcis a constant andfis differentiable at allx, then

d

dx[c f(x)] =c

d

dx[f(x)].

In simpler notation,

(c f)′=c(f)′·c f′.

In other words, the derivative of a constant times a function is equal to the constant times the derivative of the
function.
Example 3:

d

dx[^4 x

(^3) ] = 4 d
dx[x
(^3) ] = 4 [ 3 x (^2) ] = 12 x (^2).
Example 4:
d
dx

[− 2

x^4

]

=dxd[− 2 x−^4 ] =− (^2) dxd[x−^4 ] =− 2 [− 4 x−^4 −^1 ] =− 2 [− 4 x−^5 ] = 8 x−^5 =x^85.
Derivatives of Sums and Differences
Iffandgare two differentiable functions atx, then
d
dx[f(x)+g(x)] =
d
dx[f(x)]+
d
dx[g(x)]