1. Rules for Differentiation
Calculus Topics
Rules for Differentiation

2. The following symbols indicate the derivative of a function y=f(x)
The following symbols indicate the derivative of a function y=f(x). THEY ALL MEAN THE SAME THING!
Read as “y prime”
“f prime”
“dy dx” or “the derivative of y with respect to x”
“df dx”
“d dx of f at x” or “the derivative of f at x”

3. Let’s learn some shortcuts for finding derivatives!
Power Rule: If , then
Ex: If g(x)=x5, then g’(x)=5x4 .
Ex: If h(x)=x-2, then h’(x)=-2x-3 .
Constant Multiple Rule:
Ex: If f(x)=3x2, then
Sum and Difference Rule:
Ex: If f(x)= x2+x , then

4. Derivative of a Constant: If f(x)=c, then f’(x)=0.
Ex: Find the derivative of the following functions.

5. Ex: Find the derivative of the following functions.

6. Ex: Find the equation of the tangent line to f(x)=3x2+4x-2 at x=1.
We need a point and a slope…
To find slope, evaluate the derivative at x=1!
Now make an equation using (1,f(1)) as your point!

7. Ex: Find the x-values at which the tangent line to the curve of
Ex: Find the x-values at which the tangent line to the curve of is horizontal.
Horizontal tangent line means slope=0
Slope=0 means derivative=0, so find the zeros of the derivative! So

8. Product Rule:
When two functions are multiplied together, we must use product rule to find the total derivative.
Ex: Find the derivative of f(x)=(3x2-x)(4–x3).
Separate the problem into two products, u=3x2-x and v=4-x3 ...
Would you get the same answer if you multiplied out f(x) at the start, then differentiated?
YES!

9. Ex: Find the derivative of the following functions using the product rule.

10. Quotient Rule: Low d high less high d low...all over low-low!
Ex: Find the derivative of f(x).
Let u=3x3-2 and v=2x
Would you get the same answer if you separated f into two fractions?
YES!

11. Ex: Find the derivative of the following functions using the quotient rule.