Test 2 Flashcards
(f of g) ‘
( f of g)= f (g) + g (f) `
(N) / (D) `
(DN- DN) / D^2
The derivative of a constant is
0
Power rule
nx^n-1
Constant multiple Rule
d/dx cf(x) = c d/dx f(x)
Derivative of a sum
d/dx f(x) + d/dx g(x)
derivative of a difference
d/dx f(x) - g(x) = d/dx f(x) - d/dx g(x)
When finding the equation of a tangent line at x = c with y’
Differentiate the y function and then plug in c for x to find slope, then put into point slope form y-y1 = m(x-x1)
where tangent line is horizontal
m = 0
if f(x) = a^x then f’(x) = f’ (0)a^x
says the rate of change of an exponential function is proportional to the function itself
Definition of number e
such that lim h approaches 0 (e^h - 1)/h = 1
d/dx e^x
e^x
The normal line to the curve at P
The normal line at P is perpendicular to the tangent line at P (point of tangency) and slopes are opposite reciprocal)
When finding equations of the tangent line and the normal line to a curve
1) find slope of tangent line by plugging in the x value given and putting it in point slope form, then switch m’s reciprocal to find equation of tangent line
y =
mx +b
lim theta approaches 0 of (sin theta)/theta =
1
The product rule
FS’ + SF’
lim theta approaches 0 (costheta - 1)/theta =
0
Derivative of sin x
cos x
Derivative of cos x
- sin x
Derivative of tan x
sec^2 x
Derivative of csc x
-csc x cot x
Derivative of sec x
sec x tan x
Derivative of cot x
-csc^2 x
The chain rule
F ‘ (x) = f ‘ (g(x)) times g ‘ (x). Basically, derivative of the outside function times the derivative of the inside function
d/dx (a^x) =
a^x ln a
Implicit differentiation
We differentiate both sides of the equation with respect to our independent variable x, bearing in mind that y is implicitly definied as a fuction of x
Derivative of (sin^-1 x)
1/ (sqrt of 1 -x^2)
Derivative of (cos^-1 x)
-1/ (sqrt of 1-x^2
Derivative of (csc^-1 x)
-1/ x(sqrt of x^2 -1)
Derivative of (sec^-1 x)
1/ x(sqrt of x^2-1)
Derivative of (cot^-1 x)
-1/ 1 + x^2
d/dx (log a x)
1/ x ln a
d/dx (ln x)
1/x
d/dx (log a u)
1/u ln a times u’
d/dx (ln u)
1/ u(x) times u’ (x)
d/dx ln |x|
1/x
Exponential growth
if K > 0
Exponential decay
if K < 0
Differential equation for the Law of Natural growth
K is the relative growth rate, C is initial value, dy/dt = Ce^kt
Newton’s Law of Cooling
T - Ts = Ce^kt, T(t) = Ts + Ce^kt, C = T - Ts
d/dx (sinh x)
cosh x
d/dx (cosh x)
sinh x
exponential growth
dy/dx = Ce^kt
Newton’s Law of cooling
T-Ts = Ce^kt
Sinh x
e^x - e^-x
/2
cosh x
e^x + e^-x
/2
d/dx sinh x
cosh x
d/dx cosh x
sinh x
Linearization
L(x) = f(a) + f(a) (x-a)
