# How do you differentiate f(x)=lnx/sinx using the quotient rule?

Dec 8, 2015

$\frac{d}{\mathrm{dx}} \left[\frac{\ln x}{\sin x}\right] = \frac{\sin x \cdot \frac{1}{x} - \ln x \cos x}{{\sin}^{2} x}$

#### Explanation:

The quotient rule states that the derivative of a quotient of 2 functions is the denominator times the derivative of the numerator, minus the numerator times the derivative of the denominator, all divided by the square of the denominator.

ie. $\frac{d}{\mathrm{dx}} \left[\frac{f \left(x\right)}{g \left(x\right)}\right] = \frac{g \left(x\right) \cdot f ' \left(x\right) - f \left(x\right) \cdot g ' \left(x\right)}{{\left[g \left(x\right)\right]}^{2}}$

Applying this rule in this particular function gives the following result

$\frac{d}{\mathrm{dx}} \left[\frac{\ln x}{\sin x}\right] = \frac{\sin x \cdot \frac{1}{x} - \ln x \cos x}{{\sin}^{2} x}$