Derivative formulas. September 29, Derivative formulas

Transcription

1 September 29, 2013

2 Derivative of a constant function Derivative is the slope of the graph. The graph of a constant function is a horizontal line with the slope 0 everywhere.

3 Derivative of a constant function Derivative is the slope of the graph. The graph of a constant function is a horizontal line with the slope 0 everywhere. If f (x) = k, then f (x) = 0.

4 Derivative of a constant function Derivative is the slope of the graph. The graph of a constant function is a horizontal line with the slope 0 everywhere. If f (x) = k, then f (x) = 0. Example x (9) = (3) = 0. x

5 Derivative of a linear function Derivative is the slope of the graph. The graph of a linear function is a line, with a fixe slope.

6 Derivative of a linear function Derivative is the slope of the graph. The graph of a linear function is a line, with a fixe slope. If f (x) = mx + b, then f (x) = m.

7 Derivative of a linear function Derivative is the slope of the graph. The graph of a linear function is a line, with a fixe slope. If f (x) = mx + b, then f (x) = m. Example (4x + 6) = 4. x

8 Linear properties of erivative Theorem Let f an g be two functions, an a an b two constants. x [cf (x)] = cf (x)

9 Linear properties of erivative Theorem Let f an g be two functions, an a an b two constants. x [cf (x)] = cf (x) x [f (x) + g(x)] = f (x) + g (x)

10 Linear properties of erivative Theorem Let f an g be two functions, an a an b two constants. x [cf (x)] = cf (x) x [f (x) + g(x)] = f (x) + g (x) x [f (x) g(x)] = f (x) g (x)

11 Linear properties of erivative Theorem Let f an g be two functions, an a an b two constants. x [cf (x)] = cf (x) x [f (x) + g(x)] = f (x) + g (x) x [f (x) g(x)] = f (x) g (x) x [af (x) + bg(x)] = af (x) + bg (x)

12 Power rule For any constant real number n, x (x n ) = nx n 1.

13 Example Fin the erivative of the following functions:

14 Example Fin the erivative of the following functions: f (x) = 1. x

15 Example Fin the erivative of the following functions: f (x) = 1. x P(t) = t2 3 t.

16 Example Fin the erivative of the following functions: f (x) = 1. x P(t) = t2 3 t. g(w) = w

17 Example Fin the erivative of the following functions: f (x) = 1. x P(t) = t2 3 t. g(w) = w Q(x) = x 4 + 3x 2 x + 1.

18 Example Fin the erivative of the following functions: f (x) = 1. x P(t) = t2 3 t. g(w) = w Q(x) = x 4 + 3x 2 x + 1. h(θ) = θ 2 ( (θ) + 4θ 6 ).

19 Using the erivative formulas Fin the tangent line at x = 2 to the graph of y = x 4 + 2x

20 Using the erivative formulas Fin the tangent line at x = 2 to the graph of y = x 4 + 2x Slope of the line=?

21 Using the erivative formulas Fin the tangent line at x = 2 to the graph of y = x 4 + 2x Slope of the line=? Given point on the line=?

22 Example Fin an interpret the secon erivative of f (x) = x x.

23 Example The revenue (in ollars) from proucing q units of a prouct is given by R(q) = 2500q 5q 2 Fin an interpret R(100) an R (100).

24 Exponential rule For any positive constant a x (ax ) = (ln a)a x

25 Exponential rule For any positive constant a x (ax ) = (ln a)a x x (akx ) = k(ln a)a kx

26 Exponential rule For any positive constant a x (ax ) = (ln a)a x x (akx ) = k(ln a)a kx x (ekx ) = ke kx

27 Examples Fin the erivative of y = 5 + 4x 2 + e 0.02x.

28 Examples Fin the erivative of y = 5 + 4x 2 + e 0.02x. P = x x + 4e 4x

29 Derivative of ln x Theorem x (ln x) = 1 x

30 Examples Fin the erivative of y = 5 ln x x + 4e 2x.

31 Examples Suppose $1000 is eposite into a bank account that pays 8% annual interest, compoune continuously. Fin f (10) an f (10).

32 Examples Suppose $1000 is eposite into a bank account that pays 8% annual interest, compoune continuously. Fin f (10) an f (10). Explain what your answers mean in terms of money.