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More on Derivatives and Integrals -Product Rule -Chain RuleAP Physics C Mrs. Coyle
Derivative f’ (x) = lim f(x + h) - f(x ) h 0 h
Derivative Notations f’ (x) df (x) dx . f df dx
Notations when evaluating the derivative at x=af(a) df (a) dx f’(a) df |x=a dx
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
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.
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’
Example of Product Rule:Differentiate: F=(3x-2)(x2 + 5x + 1) Answer: F’(x) = 9x2 + 26x-7
If y=f(u) and u=g(x): dy = dy du dx du dxChain Rule If y=f(u) and u=g(x): dy = dy du dx du dx
Example of Chain Rule Differentiate: F(x)= (x 2 + 1) 3 Ans:F’(x)= 6(x2 +1)2x
Second Derivative Notationsdf’ (x) dx d2f (x) d x2 f’’(x)
Example of Second DerivativeCompute the second derivative of y=(x)1/2 Ans: (-1/4) x-3/2
Derivatives of Trig Functionsd sinx = cosx dx d cosx = -sinx d tanx = sec2 x dx d secx = secx tanx
Derivative of the Exponential Functiond e u = e u du dx dx
Example of derivative of Exponential Function2 Differentiate: e x Ans: 2x e x
Derivative of Ln d (lnx) = 1/x dx
a∫b f(x) dx= F(b)-F(a)= F(x)|aDefinite Integral b a∫b f(x) dx= F(b)-F(a)= F(x)|a a and b are the limits of integration.
If F(x)= ∫ f(x) dx then d F(x) = f(x) dx
Properties of Integralsa∫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
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|
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
To evaluate integrals of products of functions :Chain Rule Integration by parts Change of Variable Formula
Change of Variable FormulaWhen a function and its derivative appear in the integral: a∫b f[g(x)]g’(x) dx = g(a)∫g(b) f(y) dy
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:
Example of Change of Variable FormulaEvaluate: 0∫1 2x (x2 + 1) 9 dx Answ: 102.3
Integration by Parts a∫b u(x) dv dx= dx b = u(x) v(x)|a - a∫b v(x) du dx dx
Integration by Parts b a∫b u v’ dx= u v|a - a∫b v u’ dx
Example of Integration by PartsCompute 0∫π x sinx dx Ans: π
The Chain Rule Section 3.6c.
Calculus Final Exam Review By: Bryant Nelson. Common Trigonometric Values x-value0π/6π/3π/22π/35π/6π7π/64π/33π/25π/311π/62π sin(x)0½1½0-½-½0 cos(x)1½0-½-½0½1.
Exam 2 Fall 2012 d dx (cot x) = - csc2 x d dx (csc x) = - csc x cot x
Derivatives Part A. Review of Basic Rules f(x)=xf`(x)=1 f(x)=kx f`(x)= k f(x)=kx n f`(x)= (k*n)x (n-1) 1.) The derivative of a variable is 1. 2.)
11 The student will learn about: §4.3 Integration by Substitution. integration by substitution. differentials, and.
Clicker Question 1 What is the instantaneous rate of change of f (x ) = sin(x) / x when x = /2 ? A. 2/ B. 0 C. (x cos(x) – sin(x)) / x 2 D. – 4 /
Key Ideas about Derivatives (3/20/09)
Winter wk 6 – Tues.8.Feb.05 Calculus Ch.3 review:
4.9 Antiderivatives Wed Feb 4 Do Now Find the derivative of each function 1) 2)
CHAPTER Continuity CHAPTER Derivatives of Polynomials and Exponential Functions.
Sheng-Fang Huang. Definition of Derivative The Basic Concept.
8.2 Integration By Parts.
Differentiation Safdar Alam. Table Of Contents Chain Rules Product/Quotient Rules Trig Implicit Logarithmic/Exponential.
By: Kelley Borgard Block 4A
Aim: Differentiating Natural Log Function Course: Calculus Do Now: Aim: How do we differentiate the natural logarithmic function? Power Rule.
5.5 Bases Other Than e and Applications
Derivatives of Logarithmic Functions
Do Now Find the derivative of each 1) (use product rule) 2)
Clicker Question 1 What is the derivative of f(x) = 7x 4 + e x sin(x)? – A. 28x 3 + e x cos(x) – B. 28x 3 – e x cos(x) – C. 28x 3 + e x (cos(x) + sin(x))
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