derivative of y=logeX (1 Viewer)

[red802]

can someone help us with this problem

find the equiation of the tangent to the curve y = Inx

When x = 3

also another question, can someone show me the working our for this question

At what point on the curve y = In2x is the gradient 1/2? Find the equation of the tangent at this point

Please show working out

 

[red802]

Also, i get confused when to use In and log

 

[Riviet]

find the equiation of the tangent to the curve y = Inx

When x = 3

dy/dx=1/x
When x=3, dy/dx=1/3 and y=ln3
.'. gradient of tangent at x=3 is 1/3
Use point gradient formula: y-y₁=m(x-x₁)
y-ln3=1/3.(x-3)
y=x/3 + ln3 - 1 or x-3y+ln3-3=0

 

[Riviet]

At what point on the curve y = In2x is the gradient 1/2? Find the equation of the tangent at this point

dy/dx=2 x 1/x, by the chain rule
=2/x
when m=1/2, ie dy/dx=1/2
1/2=1/x
.'. x=2, y=ln4
Use point gradient formula again:
y-ln4=1/2.(x-2)
y=x/2+ln4-1

Also, i get confused when to use In and log

log is base 10 and can therefore be written as log₁₀. ln is "natural log", which is base e and can also be written as log_e. You will be using ln alot more of the time, especially in the logarithms/exponentials topic.

 

[pLuvia]

Both the derivatives of ln and log are 1/x

 

[_ShiFTy_]

Generally, the derivative of ln(f(x)) is f'(x)/f(x)

 

[SoulSearcher]

Another derivative of a log function, although rarely asked is:

d/dx (log_a x) = 1 / (x * log_e a), where a is any integer.

Watch out for that one, it can be asked, I saw it on the CSSA 2 Unit Trial HSC paper last year.