Derivatives The Easy Way
 Page ID
 623
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Constant Rule and Power Rule
We have seen the following derivatives:
 If f(x) = c, then f '(x) = 0
 If f(x) = x, then f '(x) = 1
 If f(x) = x^{2}, then f '(x) = 2x
 If f(x) = x^{3}, then f '(x) = 3x^{2}
 Iff(x) = x^{4}, thenf '(x) = 4x^{3}
This leads us the guess the following theorem.
Theorem d 
Proof:
We have
Applications
Example
Find the derivatives of the following functions:

f(x) = 4x^{3}  2x^{100 }

f(x) = 3x^{5} + 4x^{8}  x + 2

f(x) = (x^{3}  2)^{2 }
Solution
We use our new derivative rules to find

12x^{2}  200x^{99 }

15x^{3}+32x^{7}1

First we FOIL to get
[x^{6 } 4x^{3 }+ 4] '
Now use the derivative rule for powers
6x^{5}  12x^{2}
Example:
Find the equation to the tangent line to
y = 3x^{3}  x + 4
at the point(1,6)
Solution:
y' = 9x^{2}  1
at x = 1 this is 8. Using the pointslope equation for the line gives
y  6 = 8(x  1)
or
y = 8x  2
Example:
Find the points where the tangent line to
y = x^{3}  3x^{2 } 24x + 3
is horizontal.
Solution:
We find
y' = 3x^{2}  6x  24
The tangent line will be horizontal when its slope is zero, that is, the derivative is zero. Setting the derivative equal to zero gives:
3x^{2}  6x  24 = 0
or
x^{2}  2x  8 = 0
or
(x  4)(x + 2) = 0
so that
x = 4 or x = 2
Derivative of f(x) = sin(x)
Theorem d

Proof:
d/dx cos(x)
Theorem d

Larry Green (Lake Tahoe Community College)