Derivative of Natural Log?

Question

How do I find the derivative of Ln(5x)?

I thought the integral was the antiderivative....

If F(x) = Ln(5x) and F'(x) = 1/x, the integral of 1/x certainly doesn't become Ln(5x)...... Whats happening here?

Answer 1

If f(x) = ln x, then

f '(x) = 1/x

Here is the proof for it

y = ln x

then

ey = x

Now implicitly take the derivative of both sides with respect to x remembering to multiply by dy/dx on the left hand side since it is given in terms of y not x.

ey dy/dx = 1

From the inverse definition, we can substitute x in for ey to get

x dy/dx = 1

Finally, divide by x to get

dy/dx = 1/x

Hence proved.

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So, the derivative of Ln(5x) is 1/5x * x = 1/5

Answer 2

Derivative Of Natural Log

Answer 3

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Let's make life easy and apply the log laws. Check out this piece of magic. G(u) = ln [(3u+6)/(3u-6)]^(1/2) = 1/2 * ln [(3u+6)/(3u-6)] = 1/2 * [ln(3u+6) - ln(3u-6) ] Now differentiating G'(u) = 1/2 * [ 3/(3u+6) - 3 / (3u-6) ] = 1/2 * [1/(u+2) - 1/(u-2) ] You could leave it like that, or get a common denom G' = -2 / (u^2 - 4) = 2 / (4 - u^2)

Answer 4

As others have said, the derivative of ln(5x) = 1/x, you also ask...

"If F(x) = Ln(5x) and F'(x) = 1/x, the integral of 1/x certainly doesn't become Ln(5x)...... Whats happening here?"

that is why you add a constsnt after you perform indefinite integrals. As someone else showed, ln(5x) = ln(x) + ln(5)

so the integral of 1/x = ln(x) + C

Answer 5

The derivative of ln(5x) = 1/5x * 5(derivateive of 5x) = 1/x

Answer 6

D[ln(5x)] = 1/x

ln(5x) = ln(5) + ln(x)
the derivative of the first term is 0 because it is constant.
the derivative of the second term is 1/x.

another method:
using chain rule
D[ln(5x)] = ln'(5x) * (5x)'
= (1/5x) * 5 = 1/x

Answer 7

The derivative of ln(x) is 1/x
The derivative of ln(5x) is (1/(5x)).5=1/x.

Answer 8

ln(5x) = ln5 + lnx

ln 5 is a constant

taking the derivative we get

d(ln(5x)) / dx = d(ln5) / dx + d(lnx) / dx
= 0 + 1/x = 1/x

the antiderivative of 1/x is lnx +c

Answer 9

d / dx (ln 5x ) = 5/5x =1/x

Answer 10

Let y = ln u
u = 5x
du/dx = 5
dy/du = 1 / u
dy/dx = (dy/du) X (du/dx)
dy/dx = (1/u) X 5
dy/dx = (1 / 5x) X 5
dy/dx = 1 / x

Answer 11

chain rule
d(ln5x)/dx
= d(5x)/dx * d(ln5x)/d(5x)
= 5 * 1/5x
=5/5x
=1/x

the end
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