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More on Derivatives and Integrals -Product Rule -Chain Rule

Learn about derivatives, integrals, and the product rule, chain rule, and more in AP Physics with Mrs. Coyle. Understand derivative notations, basic derivatives, second derivatives, trig functions, exponential functions, and integrals.

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More on Derivatives and Integrals -Product Rule -Chain Rule

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  1. More on Derivatives and Integrals-Product Rule-Chain Rule AP Physics C Mrs. Coyle

  2. Derivative f’ (x) = limf(x + h) - f(x ) h0h

  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) dxdxdx or (uv)’ = u v’ + vu’

  8. Example of Product Rule: Differentiate: F=(3x-2)(x2 + 5x + 1) Answer: F’(x) = 9x2 + 26x-7

  9. Chain Rule If y=f(u) and u=g(x): dy = dydu 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) dx2 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 dsinx = cosx dx dcosx = -sinx dx dtanx = sec2 x dx dsecx = secxtanx dx

  14. Derivative of the Exponential Function d e u = e u du dxdx

  15. Example of derivative of Exponential Function 2 Differentiate: ex 2 Ans: 2x e x

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

  17. 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∫c f(x) dx =a∫b f(x) dx+b∫c f(x) dx a<b<c a∫bcf(x) dx =ca∫b f(x) dx • a∫b (f(x)+g(x)) dx =a∫bf(x) dx+ a∫b g(x) dx

  20. Basic Integrals (integration constant ommited) ∫ xndx = 1 xn+1 , n ≠ 1 n+1 ∫ exdx = ex ∫ (1/x) dx = ln|x| ∫ cosxdx = sinx ∫ sinxdx = -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) dvdx= dx b = u(x) v(x)|a - a∫bv(x) du dx dx

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

  28. Example of Integration by Parts Compute 0∫π x sinx dx Ans: π

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