1. 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)=x 5, then g’(x)=5x 4. –Ex: If h(x)=x -2, then h’(x)=-2x -3. Constant Multiple Rule: –Ex: If f(x)=3x 2, then Sum and Difference Rule: –Ex: If f(x)= x 2 +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)=3x 2 +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 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)=(3x 2 -x)(4–x 3 ). –Separate the problem into two products, u=3x 2 -x and v=4-x 3... –Would you get the same answer if you multiplied out f(x) at the start, then differentiated? YES!

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

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

11. 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!