Find dy/dx for y = x^2 / tan x.

Step-by-step solution

1. 1

   Identify the quotient rule pieces. Write $y=\frac{u}{v}$ with $u=x^2$ and $v=\tan x$. The quotient rule says $y'=\frac{u'v-uv'}{v^2}$; the order of the two products matters, so keep $u'v$ first.

2. 2

   Differentiate the numerator. By the power rule, $u'=2x$.

3. 3

   Differentiate the denominator. The standard derivative of tangent is $v'=\sec^2 x$, which itself follows from $\tan x=\frac{\sin x}{\cos x}$ and one application of the quotient rule.

4. 4

   Assemble the rule. Substituting gives $\frac{dy}{dx}=\frac{2x\tan x-x^2\sec^2 x}{\tan^2 x}$.

5. 5

   Simplify with sine and cosine. Replacing $\tan x=\frac{\sin x}{\cos x}$ and $\sec x=\frac{1}{\cos x}$ and clearing the compound fraction turns the result into $\frac{2x\sin x\cos x-x^2}{\sin^2 x}$, which is easier to evaluate numerically.

6. 6

   Note the domain. The formula holds wherever $\tan x$ is defined and nonzero, that is $x\neq\frac{\pi}{2}+k\pi$ and $x\neq k\pi$.