Problem of the DayLet f be a differentiable function such that f(3) = 2 and f '(3) = 5. If the tangent line to the graph of f at x = 3 is used to find an approximation to a zero of f, that approximation is

A) 0.4B) 0.5C) 2.6D) 3.4E) 5.5

Problem of the DayLet f be a differentiable function such that f(3) = 2 and f '(3) = 5. If the tangent line to the graph of f at x = 3 is used to find an approximation to a zero of f, that approximation is

A) 0.4B) 0.5C) 2.6D) 3.4E) 5.5

Point (3, 2)Slope = 5Tangent y - 2 = 5(x - 3)Thus y = 5x - 13To find zero 0 = 5x - 13 x = 2.6

If h(x) = f(x)g(x) what is the derivative?

If h(x) = f(x)g(x) what is the derivative?

f(x + Δx)g(x + Δx) - f(x)g(x)Δx

add a well chosen zero

f(x+Δx)g(x+Δx) + f(x+Δx)g(x) - f(x+Δx)g(x) - f(x)g(x)Δx

lim Δx 0

If h(x) = f(x)g(x) what is the derivative?

f(x + Δx)g(x + Δx) - f(x)g(x)Δx

add a well chosen zero

f(x+Δx)g(x+Δx) - f(x+Δx)g(x) + f(x+Δx)g(x) - f(x)g(x) Δx

f(x+Δx)(g(x+Δx) - g(x)) + g(x)(f(x+Δx) - f(x)) Δx

lim Δx 0

lim Δx 0

If h(x) = f(x)g(x) what is the derivative?

f(x + Δx)g(x + Δx) - f(x)g(x)Δx

add a well chosen zero

f(x+Δx)g(x+Δx) - f(x+Δx)g(x) + f(x+Δx)g(x) - f(x)g(x) Δx

f(x+Δx)(g(x+Δx) - g(x)) + g(x)(f(x+Δx) - f(x)) Δx

lim Δx 0

lim Δx 0

f(x+Δx)(g(x+Δx) - g(x)) Δx

lim Δx 0

+ g(x)(f(x+Δx) - f(x))Δx

lim Δx 0

If h(x) = f(x)g(x) what is the derivative?

f(x+Δx) (g(x+Δx) - g(x)) Δx

lim Δx 0

+ g(x) (f(x+Δx) - f(x))Δx

lim Δx 0

Evaluate limits

f(x) g'(x) + g(x) f '(x)

If h(x) = f(x)g(x) what is the derivative?

f(x) g'(x) + g(x) f '(x)

1st times derivative of second + 2nd times derivative of

1st

Product Rule

(Rule extends to cover more than 2 factors)if j(x) = f(x)g(x)h(x) then

j'(x) = f '(x)g(x)h(x) + f(x)g'(x)h(x) + f(x)g(x)h'(x)

Find the derivative of y = 2xcos x - 2sin x

Find the derivative of y = 2xcos x - 2sin x

y' = 2x(-sinx) + cos x(2) - 2cos x

y' = -2xsin x

Quotient Rule(Proof is in textbook)

if h(x) = f(x) g(x)

then h'(x) = g(x)f '(x) - f(x)g'(x) (g(x))2

Quotient Rule(Proof is in textbook)

if h(x) = f(x) g(x)

HiLo

then h'(x) = g(x)f '(x) - f(x)g'(x) (g(x))2

Lo d Hi - Hi d Lo Lo Lo

Quotient Rule

Find the derivative of 7x2 - 4 3x4 - 2x

Quotient Rule

Find the derivative of 7x2 - 4 3x4 - 2x

(3x4 - 2x)(14x) - (7x2 - 4)(12x3 - 2)

(3x4 - 2x)2

Find the derivative of x2 + 3x 6

Find the derivative of 9 5x2

Find the derivative of -3(3x - 2x2) 7x

Caution! Not every quotient needs the quotient rule.

Find the derivative of x2 + 3x 6

y = 1 (x2 + 3x) 6

y' = 1 (2x + 3) 6

Find the derivative of 9 5x2

y = 9 x-2

5

y' = 9 (-2x-3) 5

y' = -18 5x3

Find the derivative of -3(3x - 2x2) 7x

y = -3(3 - 2x) 7

y' = -3(-2) 7

y' = -6 7

Trig Derivativesd tan x = sec2x dx

d sec x = secx tanx dx

d csc x = - cscx cotx dx

d cot x = - csc2x dx

Find the derivative of y = x2csc x

Find the derivative of y = x2csc x

y' = x2(-cscx cotx) + 2x(csc x) = (x cscx)(-x cotx + 2)

How do you know when you have finished?

Combine like terms and remove negative exponents.