How do you find the derivative of y= log _ 10 x/x?

1 Answer
Jan 11, 2016

dy/dx = (1-ln(x))/(x^2ln(10))

Step by step explanation is given below

Explanation:

Interesting question! To find derivative of y=log_10(x)/x

Most people would be confused because of the log_10(x)

We know d/dx(ln(x)) = 1/x

So let us make log_10(x) into something which we know.

Here the change of base of rule would come in handy.

color(red)("Change of base rule" quad log_b(a) = ln(a)/ln(b)

Now using this with our log_10(x)
We can write it as ln(x)/ln(10) This we can work with.

Our y=log_10(x)/x

y=ln(x)/(ln(10)x)

We can use the quotient rule to simplify this.

color(red)("Quotient rule :" (u/v)'=(vu'-v'u)/v^2

"Let "u=ln(x)/ln(10) quad " and "quad quad quad v=x

Differentiating with respect to x

u' = 1/(xln(10)) " and " v' = 1

Now we find the derivative

dy/dx = (x(1/(xln(10))-ln(x)/ln(10)*1)) /x^2

dy/dx = (1/ln(10)-ln(x)/ln(10)) /x^2

dy/dx = (1-ln(x))/(x^2ln(10))