Differentiation rule - ch 3 Flashcards
What is the normal line?
perpendicular to tangent line
slope = negative reciprocal
Is this true
Derivative of a constant times a function is the constant times derivative of the function?
True.
Constant pops out.
What’s the definition and significance of the number e?
e^x is its own derivative.
Slope of y=e^x is equal to its y coordinate of the point (p 179)
What type of real-world applications are trig functions used for?
quantities that vary in periodic manner
e.g.
vibrations, waves, elastic motions
What’s the product rule of derivatives?
What’s the quotient rule of differentiation?
Table of differentiation formulas to know
Derivatives of trig functions
The trig functions that start with letter “c” have negative derivative
what is derivative of a^x
(a^x) (ln a)
So f’(e^x) = e^x.
= repeat * ln base
Is e^x the only function that is its own derivative?
No
y=0
lim (x –> 0) sin(h) / h =
1
What is the Chain Rule?
if g is differentiable at x and f is differentiable at g(x) then f(g(x)) is differentiable at x and …
F’(x) = f’(g(x))*g’(x)
dy/dx = dy/du * du/dx
Deriv outer of inner times deriv inner
Derivative of sin(cos(tan))
Chai rule twice
Find f’(x) if f=sqrt (x^2 + 1)
Pic. Solve before changing original equation
Thus dy/dx = - x/y
Solve first change equation setting y one side
Y’’ if x^4 + y^4 = 16
Second derivative
Then substitute y’ in and then original equation to get
Y’’= -48 x^2\y^7
Pg 212
What does acrsin function mean?
Sin (y) = x
d/dx (logb x)
exponentiate to get the b^y
Remember this!
d/dx ln x
1/x
b/c ln e = 1
d/dx ln g(x)
g’(x) / g(x)
d/dx f(x) = ln (|x|)
pic
pic
Why use “implicit differentiation”?
On graphs that fail the vertical line test, like circles (aka implicit functions where you cannot solve for y explicitly)
x^2 + y^2 = 1
pic
Can replace coxy to have function of only x… use scocahtoa in top right
differentiate arctan
Notice: variable ^ variable
Then use logarithmic differentiation
What is the limit definition of e?
d/dx b^g(x)
b^g(x) * ln(b) * g’(x)
