Hello and Welcome. Suppose I have the function f(x) = (x-1)ln(x)...
and I want find the...
the derivative of this function. Well, first of all I can realize that...
this function is composed of two parts:
the first part I'll call...
u(x), which is a sub-function...
and the second part...
I'll call v(x)...
and it's another function. So, this means...
f(x) is the product of two...
functions: u and v... and I'll put a dot here to...
emphasize that it's a product of two functions
the function u(x) and the function v(x) and therefore...
to differentiate this, I can use the product rule...
and the product rule states that the differential of f(x)...
is equal to...
v(x), the second part untouched, multiplied by...
the differential of u(x), plus...
u(x), the first part untouched, multiplied by the differential v(x)
So since...
in my case, u = x - 1
that implies then the differential of u...
with respect to x is equal to 1
x differentiates to 1, and the -1 as a constant differentiates to 0...
and my function v(x)...
which is equal to ln(x)...
well that differentiates to...
1 / x
So I can simply now plug these into the product rule to get...
the differential of f(x). So the differential of f(x) should equal then...
v(x), which is...
ln(x) multiplied by...
1, plus...
(x - 1)
multiplied by 1/x and...
to finish off, I can write this more neatly as...
ln(x) + 1 - 1/x
So what I've done here is...
multiplied the 1/x into these brackets...
and that's pretty much all there is to using
the product rule.
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