Vocab. and Properties
Day 1 Problems
Day 2 Problems
Day 3 Problems
Mystery
150

What is the properties of logarithms (logbMN, logb(M/N), logbMr)?

logb(MN)=logbM+logbN

logb(M/N)=logbM-logbN

logbMr=rlogbM

150

find d/dx(xlnx)

y'=(x/x)+lnx=1+lnx

Must use the product rule.

150

Find the derivative of y=x^x using Log. Functions

lny=xlnx

dy/dx(1/y)=x(1/x)+lnx

dy/dx=y(1+lnx)

dy/dx=(x^x)(1+lnx)

150

Find d/dx(log5x)

 f'(x)=1/(xln5)

150

Find the derivative of the function j(x) when j(x)=xlnx-x

j'(x)=x(1/x)+lnx-1

=1-1+lnx

=lnx

200

What is the derivative (dy/dx) of each basic log function:

f(x)=lnx, g(x)=ln|x|, h(x)=lnu

f'(x)=1/x

g'(x)=1/x

h'(x)=(1/u)(du/dx)

200

Find the derivative of m(x)=ln(x^2+1)

M'(x)=(2x)/(x^2+1) use ln rules and take derivative

200

Differentiate y=(3x^2+5)^(1/x)

lny=(1/x)ln(3x^2+5) take natural log

y'/y=(-1/x^2)ln(3x^2+5)+(1/x)(6x/(3x^2+5))

use implicit differentiation

y'=(3x^2+5)^(1/x)[(-1/x^2)ln(3x^2+5)+(1/x)(6x/(3x^2+5)] simpilify by multiplying by y

200

find the derivative of y=e^(sinx)

y=sinxlne

y'/y=cosx

y'=cosx(e^(sinx))


200

Find the derivative of g(x) when g(x)=(ln(lnx))3

g'(x)=3(ln(lnx))2(1/lnx)(1/x)

g'(x)=(3(ln(lnx))2)/(xlnx)

300

What is the trick/rule for finding the derivative of log functions such as f(x)=logbu?

*find the derivative in terms of these variables*

f'(x)=(du/dx)(1/ulna)

300

find the derivative of y=ln(cosx)

y'=(1/cosx)(-sinx) take the derivative

=(-sinx)/(cosx) simplify by multiplying

=-tanx simply using knowledge of trig. functions

300

Find the derivative of y=(x-2)^(x+1) using log. functions

lny=(x+1)ln(x-2)

(1/y)(dy/dx)=(x+1)(1/(x-2))+ln(x-2)

dy/dx=y((x+1)/(x-2))+ln(x-2))

dy/dx=((x-2)^(x+1))(((x+1)/(x-2))+ln(x-2))

300

Find the derivative of y=x2e5x

y'=2xe5x+5x2e5x

300

find the derivative of y=e(lnx)^2

y'=2e(lnx)^2((lnx)/x)

400

What are these properties of Exponents?

1. am•an=

2. (am)n=

3. (ab)m=

1. am+n

2. amn

3. ambn

400

Find the derivative of g(x)=ln((x+4)/(x-3))

g(x)=ln(x+4)-ln(x-3) use properties

g'(x)=1/(x+4)-1/(x-3) take the derivative

=-7/((x+4)(x-3)) simplify

400

Find the derivative of y=(sinx)^(x^3)

lny=(x^3)(ln(sinx)) take ln of both sides

y'/y=3x^2(ln(sinx))+(cosx/sinx)(x^3) take derivative of both sides

y'=(sinx)^(x^3)[3x^2(ln(sinx))+cotx(x^3)]

multiply but y and simplify

=(sinx)^(x^3)[(x^3)cosx+(3x^2)sinxln(sinx)] simplify trig. functions more

400

Find the derivative of  y=log3(sinx)

y=(ln(sinx))/(ln3)

y'=(1/ln3)(1/sinx)(cosx)

y'=cosx/ln3

400

Find the derivative of f(x) when f(x)=log10(x+2x)

f(x)=(ln(x+2x))/(ln(10))

f'(x)=(1/ln10)((1+2xln2)/(x+2x)


500

What are these properties of Exponents?

4. am/an=

5. (a/b)m=

6. xa/b=

4. am-n

5. (am)/(an), b≠0

6. (^a√x)b

*(^a√x) just means the a root of x*

500

find the derivative of y=ln(x+3)(5x-2)(3x+1)

y=ln(x+3)+ln(5x-2)+ln(3x+1) Use properties

y'=1/(x+3)+5/(5x-2)+3/(3x+1) take the derivative

500

Find the Derivative of y=(18x)sin7x using log. functions

lny=sin7x(ln18x)

y'/y=7cos7x(ln18x)+sin7x(1/x)

y'=18xsin7x(7cos7x(ln18x)+sin7x(1/x))

500

Find the derivative of h(x)=log3((x^3)(4x+6)^1/2)/5)

h(x)=log3x+1/3log3(4x+6)-log35

h'(x)=1/xln3+4/(12xln3)-0

=(12x+18+4x)/(3x(4x+6)ln3)

=(8x+9)/((2x+3)ln3)

500

Find v'(x) when v(x)=33x+6tanx

y'=33x+6sec2x+33x+6(3ln3)tanx