1. 3.3 – Rules for Differentiation
Ch. 3 – Derivatives
3.3 – Rules for Differentiation

2. 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

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

4. To find a second derivative, find the derivative of the derivative!
Ex: Find the second derivatives of the following functions.

5. 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!

6. 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

7. 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!

8. Quotient Rule:
Low d high minus high d low...all over the square below!
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!

9. Ex: A formula for relating pressure and volume for an ideal gas is listed below. Assuming n, R, and T are constants, find dP/dV.
We are differentiating w/respect to V!
Treat n, R, and T as if they were constant numbers!