Download presentation

Published byAntoine Plater Modified over 7 years ago

1
**More on Derivatives and Integrals -Product Rule -Chain Rule**

AP Physics C Mrs. Coyle

2
Derivative f’ (x) = lim f(x + h) - f(x ) h 0 h

3
Derivative Notations f’ (x) df (x) dx . f df dx

4
**Notations when evaluating the derivative at x=a**

f(a) df (a) dx f’(a) df |x=a dx

5
Basic Derivatives d(c) = 0 dx d(mx+b) = m dx d(x n) = n x n-1 dx n is any integer x≠0

6
**Derivative of a polynomial.**

For y(x) = axn dy = a n xn-1 dx -Apply to each term of the polynomial. -Note that the derivative of constant is 0.

7
Product Rule For two functions of x: u(x) and v (x) d [u(x) v (x)] =u d v (x) + v d u (x) dx dx dx or (uv)’ = u v’ + vu’

8
**Example of Product Rule:**

Differentiate: F=(3x-2)(x2 + 5x + 1) Answer: F’(x) = 9x2 + 26x-7

9
**If y=f(u) and u=g(x): dy = dy du dx du dx**

Chain Rule If y=f(u) and u=g(x): dy = dy du dx du dx

10
Example of Chain Rule Differentiate: F(x)= (x 2 + 1) 3 Ans:F’(x)= 6(x2 +1)2x

11
**Second Derivative Notations**

df’ (x) dx d2f (x) d x2 f’’(x)

12
**Example of Second Derivative**

Compute the second derivative of y=(x)1/2 Ans: (-1/4) x-3/2

13
**Derivatives of Trig Functions**

d sinx = cosx dx d cosx = -sinx d tanx = sec2 x dx d secx = secx tanx

14
**Derivative of the Exponential Function**

d e u = e u du dx dx

15
**Example of derivative of Exponential Function**

2 Differentiate: e x Ans: 2x e x

16
Derivative of Ln d (lnx) = 1/x dx

17
**a∫b f(x) dx= F(b)-F(a)= F(x)|a**

Definite Integral b a∫b f(x) dx= F(b)-F(a)= F(x)|a a and b are the limits of integration.

18
**If F(x)= ∫ f(x) dx then d F(x) = f(x) dx **

19
**Properties of Integrals**

a∫b cf(x) dx =c a∫b f(x) dx a∫c f(x) dx = a∫b f(x) dx+ b∫c f(x) dx a<b<c a∫b (f(x)+g(x)) dx = a∫b f(x) dx+ a∫b g(x) dx

20
**Basic Integrals (integration constant ommited)**

∫ xn dx = 1 xn+1 , n ≠ 1 n+1 ∫ ex dx = ex ∫ (1/x) dx = ln|x| ∫ cosx dx = sinx ∫ sinx dx = -cosx ∫ (1/x) dx = ln|x|

21
**Example with computing work.**

There is a force of 5x2 –x +2 N pulling on an object. Compute the work done in moving it from x=1m to x=4m. Ans: 103.5N

22
**To evaluate integrals of products of functions :**

Chain Rule Integration by parts Change of Variable Formula

23
**Change of Variable Formula**

When a function and its derivative appear in the integral: a∫b f[g(x)]g’(x) dx = g(a)∫g(b) f(y) dy

24
**Example: When a function and its derivative appear in the integral:**

Compute x=0∫x=1 2x (x2 +1) 3 dx Ans: 3.75 Ans:

25
**Example of Change of Variable Formula**

Evaluate: 0∫1 2x (x2 + 1) 9 dx Answ: 102.3

26
Integration by Parts a∫b u(x) dv dx= dx b = u(x) v(x)|a - a∫b v(x) du dx dx

27
Integration by Parts b a∫b u v’ dx= u v|a - a∫b v u’ dx

28
**Example of Integration by Parts**

Compute 0∫π x sinx dx Ans: π

Similar presentations

© 2021 SlidePlayer.com Inc.

All rights reserved.

To make this website work, we log user data and share it with processors. To use this website, you must agree to our Privacy Policy, including cookie policy.

Ads by Google