Question

How do you use the quotient rule to differentiate `y=cos(x)/ln(x)`?

Answer 1

The quotient rule states:

`d/dx[f(x)/g(x)] = (f'(x)g(x) - f(x)g'(x))/((g(x))^2)`

Let `f(x) = cosx`, and let `g(x) = ln x`.

We know that the derivative of `cosx` is `-sinx`, and that the derivative of `ln x` is `1/x`. Therefore, `f'(x) = -sinx`, and `g'(x) = 1/x`.

Now we may simply plug into the quotient rule formula:

`dy/dx = (-sinxlnx - cosx/x)/((ln x)^2)`

And there is our answer. If we would like, we can split this fraction up to make it a bit prettier:

`dy/dx = -(sinxlnx)/((ln x)^2) - (cosx/x)/((ln x)^2)`

This simplifies to:

`dy/dx = -(sinx)/(ln x) - cosx/(x(ln x)^2)`