Mastering Derivative Rules: Application, Calculation, and Understanding

1. Section 2.3 Rules for Differentiation

2. Do-Now: Homework quiz • Fill in the blanks. • The formula _________(1)__________ serves as the limit definition of a derivative. By evaluating this limit at a particular value of x, you can calculate the ______(2)______ of the _______(3)________ line to a curve f(x) at x. If, for example, the function f(x) is increasing over the interval (-2, 5), the values of the function f’(x) will be _______(4)________ over that same interval.

3. Calculating dy/dx • On the following slide is a list of functions. For each one, we will calculate dy/dx using the limit definition of the derivative. • Your goal is to compare the function to its derivative and try to recognize patterns.

4. Find dy/dx for each function • 1. y = 5x 2. y = 5x + 13 • 3. y = 2x2 + 5x 4. y = 3x2 – 5x • 5. y = 1/(3x) 6. y = x3 • 7. y = 7x2 8. y = x2 – 6

5. Derivative Rules

6. Derivative Rules

7. Finding derivatives • y = x4 – 2x3 + 4x2 – 8x – 13 + 1/x • y‘ = = ? • You can also calculate higher order derivatives. • y‘’ = = ? • y‘’’ = ? • y(4) = ?

8. Product Rule • Can you find the derivative of a product the same way you would the derivative of a sum or difference? Try it with x2. • I prefer writing the product rule with the terms switched.

9. Quotient Rule

10. Product and quotient rule examples. • For each problem, find f’(x). • 1. • 2. • 3. f(x) = (4x2 – 6)(3x3 + 5) • Check to see if you get the same answer by expanding first as you do when using the product rule.

11. Application of Derivative Rules • Find the equation for the tangent line to the following function at x = 1. • Suppose that u and v are differentiable at x = 5 and that u(5) = 7, v(5) = 2, u’(5) = -3 and v’(5) = 6. • Find (a) d/dx (u/v) and (b) d/dx(10uv) at x = 5.