More on Derivatives and Integrals Product Rule Chain

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

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

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

Notations when evaluating the derivative at x=a f(a) f’(a) df (a) dx df |x=a dx

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

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

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

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

Second Derivative Notations ’ df (x) dx 2 df d x 2 f’’(x)

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

Derivatives of Trig Functions d sinx = cosx dx d tanx = sec 2 x dx d cosx = -sinx dx d secx = secx tanx dx

Derivative of the Exponential Function de dx u = e u du dx

Example of derivative of Exponential Function 2 Differentiate: 2 Ans: 2 x e x

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

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

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 5 x 2 –x +2 N pulling on an object. Compute the work done in moving it from x=1 m to x=4 m. • Ans: 103. 5 N

To evaluate integrals of products of functions : • Chain Rule • Integration by parts • Change of Variable Formula

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

Example: When a function and its derivative appear in the integral: • Compute • Ans: 3. 75 x=1 2 x (x 2 +1) 3 dx ∫ x=0

Example of Change of Variable Formula Evaluate: 1 2 x (x 2 + 1) 9 dx ∫ 0 Answ: 102. 3

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

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

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