AP Calculus BC: Derivatives

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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?

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


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