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PURDUE / Calculus, math / CAL 161 / What is the derivative of c expression?

What is the derivative of c expression? Description

Description: Study guide for Exam 2
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Calc 161 Exam 2 Study Guide

Expression

Derivative

C

0

xp

pxp-1

ex

ex

ax

Ln(a)ax

Ln(x)

1/x

Cf(x)

Cf’(x)

f(x)+g(x)

f’(x)+g’(x)

f(x)-g(x)

f’(x)-g’(x)

f(x)/g(x)

(f’(x)g(x)-g’(x)f(x))/ (g(x))2

f(x)*g(x)

f’(x)g(x)+g’(x)f(x)

f(g(x))

f’(g(x))g’(x)

Sin x

Cos x

Cos x

-sin x

Tan x

Sec2x

Csc x

-csc x cot x

Sec x

sec x tan x

Cot x

-csc2 x

What is the derivative of c expression?

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Sin-1 x

1/sqrt(1-x2)

Tan -1 x

1/(x2+1)

Sinh x

Cosh x

Cosh x

Sinh x

Tanh x

Sech2 x

Sech x

-sech x tan x

Csch x

-csch x coth x

Coth x

-csch2 x

What is the derivative of cf(x) expression?

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Power Rule:

d/dx (xp) = pxp-1 If you want to learn more check out What is the state of the evidence for emotional disclosure in the pennebaker study?

d/dx (C * f(x)) = C * d/dx (f(x))

Product Rule:

d/dx (f(x)*g(x)) = f’(x)g(x)+g’(x)f(x)

Quotient Rule:

d/dx (f(x)/g(x)) = (f’(x)g(x)-g’(x)f(x))/(g(x))2

Trig Facts:

Pi/6

Pi/4

Pi/3

Pi/2

Sin θ

½

1/sqrt(2)

Sqrt(3)/2

1

Cos θ

Sqrt(3)/2

1/sqrt(2)

½

0

Tan θ

1/sqrt(3)

1

Sqrt(3)

DNE

What is the expression of 0 derivative?

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Tan x = sin x/cos x arcsin has domain [-1,1] range [- pi/2, pi/2]

Sec x = 1/cos x arccos has domain [-1,1] range  [0,pi]

Csc x = 1/sin x arctan has domain (-Ꝏ,Ꝏ) range  (-pi/2,pi/2)

Cot x = cos x/sin x

Cos2 x +sin2 x = 1

1+ tan2 x = sec2 x We also discuss several other topics like What are the five stages of dying?
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Chain rule:

d/dx(f(x)) = f’(x)(x’)

Implicit Differentiation

X2+y2 = 1

d/dx (X2+y2) = 0

2x+2yy’=0

2yy’ = -2x

Y’ = -x/y

Logarithmic Differentiation;

d/dx (ln x) = 1/x

d/dx (loga x) = (1/ln a)(1/x)

d/dx (ln 3x) = (1/3x)(3) = 1/x

d/dx (ln 3 + ln x) = 0+1/x = 1/x

d/dx (ln x2) = d/dx (2 ln x) = 2(1/x) = 2/x

Hyperbolic functions:

Sinh x = (ex – e-x)/2

Cosh x = (ex + e-x)/2

Cosh2 x – sinh2 x =1

Growth:

Y’ = ky

Y(t) = yoekt

t = (ln 2)/k t= time to double

2=ekt => double

K = (ln 1/2 )/t1/2 => half life

Newton’s Law of Cooling

dT/dt = k(T-Ts) Assume Ts is 70 if not given Related Rates:

1. Draw picture

2. Label things that change with variables 3. Identify equations

4. Differentiate using chain rule

5. Plug n chug

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