Important Rules/Formulas
Power Rule:
f(x) = x^r
f ‘(x) = r * x^(r-1)
Example
Constant-Multiple Rule:
(K is a constant) d/dx (K * f(x) ) = K * f ‘(x)
Example
Sum Rule:
d/dx ( f(x) + g(x) ) = f ‘(x) + g ‘(x)
Example
Chain Rule:
d/dx ( (f(x)^r)) = r * f(x)^(r-1) * f ‘(x)
Example
Examples
Power Rule
f(x) = x^2
We use the Power Rule above.
We multiply the function (x^2) by the power (2) and then subtract one from the power (2-1)
f ‘(x) = 2 * x^(2-1) = 2x
Constant-Multiple Rule
f(x) = 5x^4
Using Rule #2
We just rewrite the 5, and treat the rest of the function (x^4) just like in example 1
Then once we get the answer, we multiply it by the 5 from the original problem
f ‘(x) = 5 * 4 * x^(4-1)
= 5 * 4 * x^3
= 20x^3
Sum Rule
f(x) = 6x^3 + 4x^2
Using Rule #3
We treat each part of the function individually.
First we take the derivative of 6x^3 using rules #1 and #2 and then do the same for 4x^2
Then we add the two derivatives together
f ‘(x) = 6 * 3 * x^(3-1) + 4 * 2 * x^(2-1)
= 18x^2 + 8x
Chain Rule
f(x) = (5x^2 + 3) ^2
This is a two step problem. We “ignore” the “inner” part of the function, which is (5x^2 +3)
((By ignore, I mean that we simply rewrite it without doing anything to it))
And we use rule #1, multiplying the function by the power (2) and subtracting one from the power, but we leave the inside the same.
2 * (5x^2 + 3) ^1 (The First Part)
This is NOT the answer!
Next we must now look at the “inner” part of the function and take the derivative of that
2 * 5x = 10x (The Second Part)
To get the answer, we write (The First Part) multiplied by (The Second Part)
2 * (5x^2 + 3) * 10x
= 20x * (5x^2 + 3)