Chapter 3 Derivatives
Quotient rule for derivatives: d/dx (f(x)/g(x))=
(g(x)f'(x) - f(x)g'(x))/((g(x))² =(lo d hi - hi d lo)/ (lo lo) Order matters bc we're subtracting!
Derivative of fractions like 1/x^5
1/x^5 = x^-5 f'(x) = -5x^-6
The Chain Rule
3.4 Taking derivatives of composite functions (one function inside another ex: (2x+12)^8) RULE: F(x) = f(g(x)) F'(x) = f'(g(x) * g'(x) first derivative of outside, leave inside alone multiply ^ by derivative on inside
Implicit differentiation
3.5 Example: x^3 +y^3 =6xy it's impossible to solve and get y as a function of x. In this case we use Implicit differentiation
Derivatives of log functions 3.6
3.6
second derivative (f'') tells...
Acceleration
Derivatives of trig functions
All "co" functions have - in derivative
Examples that would need chain rule
Cos(5x^3) x^4(5x^3+8x)
third derivative (f''') tells...
Jerk
Using the chain rule and product rule
Reminder- chain rule f'g * g' product rule fg' + gf' Example: f(t) = (3t-1)^4 (2t+1)^-3 f'(t) = ((3t-1)^4)*((-3(2t+1)^-4)*2) + here do g*f' ^f. ^g'.
Using the chain rule and quotient rule
Reminder- chain rule f'g * g' quotient rule (gf' - fg')/g^2 Example: S(t) = Sqrt((1+sin(t))/(1+cos(t))) what is S'(x) S(t) = (1+sin(t))/(1+cos(t))^1/2 Basically, do outside function and then inside separately. Clearer example in next slide
Visualizing derivatives
When f changes slope, f' hits 0 When f has posi slope, f' y values are posi applies for f' & f'' f'' and f''' etc
Arc(anything) = inverse
arcsin= sin^-1 etc etc
derivatives of logs
d of log base x = 1/(x*ln(b)) d of ln(x) = 1/x
Derivative of constant ex: f(x) = 4
f'(x) = 0 BC derivative = slope of tangent line. derivative os constant is always 0
f(x) = a^x a>0 and not = to 1 what is f'(x)
f'(x) = a^x * ln(a)
f(x) = e^x what is f'(x)
f'(x) = e^x
f(x) = sqrt(x)
f(x) = x^1/2 f'(x) = 1/2 * x^-3/2
Power rule
f(x) = x^n f'(x) = nx^(n-1) ex: f(x) = 2x^4 f'(x) = 4*2x^(4-1) f'(x) = 8x^3 f''(x)= 24x^2 f'''(x) 48x
Logarithm Formula: log base x = y and ln(x)
log 10 1000 = 3 What do we raise 10 to the power of to get 1000 on ti84 is under f4 or alpha trace ln is natural log
perpendicular slopes
negative reciprocals of original slope Ex: 1/2 -> -2 3/4 -> -4/3
product rule: p(x) = f(x)⋅g(x) p'(x) = ?
p'(x) = (f'(x) * g(x)) = (g'(x) *f(x)) Order doesn't matter here!
original function (f) tells...
position
Implicit differentiation rules
rules to differentiate: 1- treat x as usual 2- everytime we differentiate y, add y' 3- solve to get y' alone
Value of the derivative
slope of the tangent line
first derivative (f') tells...
velocity
Implicit differentiation example
x^3 + y^3 = 6xy 3x^2 + (3y^2)*y' = 6y + 6xy' Now, solve for y'