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

AP Physics C Mrs. Coyle

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

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

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

f(a) df (a) dx f’(a) df |x=a dx

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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

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**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.

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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’

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**Example of Product Rule:**

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

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**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

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

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**Second Derivative Notations**

df’ (x) dx d2f (x) d x2 f’’(x)

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**Example of Second Derivative**

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

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**Derivatives of Trig Functions**

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

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**Derivative of the Exponential Function**

d e u = e u du dx dx

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**Example of derivative of Exponential Function**

2 Differentiate: e x Ans: 2x e x

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

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**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.

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

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**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

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**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|

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**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

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

Chain Rule Integration by parts Change of Variable Formula

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**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

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**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:

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**Example of Change of Variable Formula**

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

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

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

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**Example of Integration by Parts**

Compute 0∫π x sinx dx Ans: π

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