What is the properties of logarithms (logbMN, logb(M/N), logbMr)?
logb(MN)=logbM+logbN
logb(M/N)=logbM-logbN
logbMr=rlogbM
find d/dx(xlnx)
y'=(x/x)+lnx=1+lnx
Must use the product rule.
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)
Find d/dx(log5x)
f'(x)=1/(xln5)
Find the derivative of the function j(x) when j(x)=xlnx-x
j'(x)=x(1/x)+lnx-1
=1-1+lnx
=lnx
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)
Find the derivative of m(x)=ln(x^2+1)
M'(x)=(2x)/(x^2+1) use ln rules and take derivative
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
find the derivative of y=e^(sinx)
y=sinxlne
y'/y=cosx
y'=cosx(e^(sinx))
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)
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)
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
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))
Find the derivative of y=x2e5x
y'=2xe5x+5x2e5x
find the derivative of y=e(lnx)^2
y'=2e(lnx)^2((lnx)/x)
What are these properties of Exponents?
1. am•an=
2. (am)n=
3. (ab)m=
1. am+n
2. amn
3. ambn
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
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
Find the derivative of y=log3(sinx)
y=(ln(sinx))/(ln3)
y'=(1/ln3)(1/sinx)(cosx)
y'=cosx/ln3
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)
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*
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
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))
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)
Find v'(x) when v(x)=33x+6tanx
y'=33x+6sec2x+33x+6(3ln3)tanx