AP Calculus BC: Derivatives
What are the three reasons why a derivative does not exist at a certain point?
Discontinuity, f(x)⁻≠f(x)⁺, vertical tangent line
When is f(x) differential at x=a?
If f'(a) exists
What is the power rule?
If f(x)=mxⁿ where m and n are constants, then f'(x)=mnxⁿ⁻¹
What is quotient rule?
If h(x)=f(x)/g(x), then h'(x)=[g(x)f'(x)-f(x)g'(x)]/(g(x))²
What is the product rule?
If h(x)=f(x)g(x), then h'(x)=f'(x)g(x)+f(x)g'(x)
Graph f(x)=³√(x-3). Is f(x) differentiable? If not, why?
No, f(x) has a vertical tangent line at x=3
Graph f(x)=1/x. Is f(x) differentiable? If not, why?
No, f(x) is not continuous at x=0
Graph f(x)=|x|. Is f(x) differentiable? If not, why?
No, f(x)⁻≠f(x)⁺
T/F: The rate of change of any exponential function is proportional to the function itself.
T
What is a derivative?
The slope of a tangent line
What is average rate of change?
The slope of the secant line that passes through two points
What is instantaneous rate of change?
The slope of the tangent line at a point
Graph f(x)=x². Is f(x) differentiable? If not, why?
Yes
What is the chain rule?
[f(g(x))]'=f'(g(x))g'(x)
What is the derivative of sin x?
cos x
d/dx[cos⁻¹(u)]=?
d/dx[cos⁻¹(u)]= -u'/√(1-u²)
d/dx[cot⁻¹(u)]=?
d/dx[cot⁻¹(u)]= -u'/u²+1
d/dx[csc⁻¹(u)]=?
d/dx[csc⁻¹(u)]= -u'/[|u|√(u²-1)]
d/dx[sec⁻¹(u)]=?
d/dx[sec⁻¹(u)]= u'/[|u|√(u²-1)]
d/dx[sin⁻¹(u)]=?
d/dx[sin⁻¹(u)]= u'/√(1-u²)
d/dx[tan⁻¹(u)]=?
d/dx[tan⁻¹(u)]= u'/u²+1
How do you write y'(x) using d?
dy/dx
How do you denote the third derivative in terms of d?
d³y/dx³
What is the derivative of eˣ?
eˣ
f(x)=xˣ, f'(x)=?
f'(x)=((ln x)+1)xˣ
f(x)=x^(2/3)-4x⁹-11x, f'(x)=?
f'(x)=(2/3)x^(-1/3)-36x⁸-11
f(x)=ln(x²+4x), f'(x)=?
f'(x)=(2x+4)/(x²+4x)
f(x)=(x²-7x+1)(x³+9x), f'(x)=?
f'(x)=(2x-7)(x³+9x)+(x²-7x+1)(3x²+9)
f(x)=(9x⁴+12x-11)(√x+8), f'(x)=?
f'(x)=(36x³+12)(√x+8)+(9x⁴+12x-11)(1/(2√x))
f(x)=10ˣ⁻⁷, f'(x)=?
f'(x)=(ln 10)10ˣ⁻⁷
f(x)=7ˣ, f'(x)=?
f'(x)=(ln 7)7ˣ
f(x)=csc(x+3), f'(x)=?
f'(x)=-csc(x+3)cot(x+3)
f(x)=ln x, f'(x)=?
f'(x)=1/x
f(x)=10eˣ⁺³, f'(x)=?
f'(x)=10eˣ⁺³
f(x)=7x⁴+3x²-9, f'(x)=?
f'(x)=28x³+6x
f(x)=x², f'(x)=?
f'(x)=2x
f(x)=t³cos t-2ᵗ⁻¹, f'(x)=?
f'(x)=3t²cos t-t³sin(t)-(ln 2)2ᵗ⁻¹
f(x)=x³, f'(x)=?
f'(x)=3x²
f(x)=(2x-1)⁴, f'(x)=?
f'(x)=8(2x-1)³
f(x)=9eˣ, f'(x)=?
f'(x)=9eˣ
f(x)=(xeˣ)/(5x⁴-3x³) f'(x)=?
f'(x)=[(5x⁴-3x³)(xeˣ+eˣ)-xeˣ(20x³-9x²)]/(5x⁴-3x³)²
f(x)=(x-3)^√x, f'(x)=?
f'(x)=[(x-3)^√x]*[[ln(x-3)/2√x]+[√x/(x-3)]]
f(x)=(x²+8x-7)/(x³+9), f'(x)=?
f'(x)=[(x³+9)(2x+8)-3x²(x²+8x-7)]/(x³+9)²
f(x)=eˢᵉᶜ⁽⁴ˣ⁻ˣ^²⁾, f'(x)=?
f'(x)=eˢᵉᶜ⁽⁴ˣ⁻ˣ^²⁾(sec(4x-x²)tan(4x-x²))(4-2x)
f and g are inverses. f=x². What is g'(2)?
g'(2)=1/2√2, -1/2√2
f and g are inverses. f=x²+x. What is g'(2)?
g'(2)=1/3, -1/3
f and g are inverses. f=-2x³-x²-9. What is g'(3)?
g'(3)=-1/20
f and g are inverses. f=2x+1. What is g'(3)?
g'(3)=1/2
f and g are inverses. f=(1/4)x³+x-1. What is g'(3)?
g'(3)=1/4
If f and g are inverses of each other, what is g'(x)?
g'(x)=1/f'[g(x)]
What is the derivative of sec x?
sec x tan x
What is the derivative of tan x?
sec²x
How do you denote the third derivative in terms of y(x)?
y'''(x)
Given y'=(2x-1)/3y, find y''.
y''=[6y²-(2x-1)²]/9y³
If y=alogᵦ(u(x)), then y'=...
y'=au'(x)/[(ln b)(u(x))]
If y=aln(u(x)), then y'=...
y'=au'(x)/u(x)
Write the tangent line of y=3x²-3x+7 when x=-2
y-25=-15(x+2)
Find all of the turn points of f(x)=x³-3x²-2
(0, -2), (2, -6)
What is the derivative of f(x)=aˣ
(ln a)aˣ
If the position of an object relative to time is x(t)=t³-3t²+7, what is the average velocity from 0 seconds to 2 seconds?
-2 u/s
6xy=x³+y³, y'=?
y'=(3x²-6y)/(6x-3y²)
What is the derivative of csc x?
-csc x cot x
What is the derivative of cot x?
-csc²x
What is the derivative of cos x?
-sin x
Find the slope of the tangent line of f(x)=eˣ/(1+x²) when x=1
0
If the position of an object relative to time is x(t)=t³-3t²+7, what is the instantaneous velocity at 2 seconds?
0
What is the slope of the tangent line at a turn point?
0
x²+y²=9, y'=?
y'=-x/y
y²=x, y'=?
y'=1/2y
y=sec⁻¹(x²), y'=?
y'=2x/[x²√(x⁴-1)]
f(x)=log₇(5x-1), f'(x)=?
y'=5/[(ln 7)(5x-1)]
sin(x+y)=y²cosx, y'=?
y'=[-cos(x+y)-y²sinx]/[cos(x+y)-2ycosx]
Differentiate y=[³√(x²-7x+4)]/[(5x-4)⁷(x+3)¹⁵] using logarithmic differentiation
y'=[³√(x²-7x+4)]/[(5x-4)⁷(x+3)¹⁵]*[[(2x-7)/3(x²-7x+4)]-[35/(5x-4)]-[15/(x+3)]]
What is d/dy(y⁷)?
7y⁶
What is d/dx(y⁷)?
7y⁶y'
What is a cusp?
A corner in the graph
What is a higher order derivative?
A derivative of a derivative
