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

1 Answer
Aug 4, 2014

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)