Find the derivative of tan(x)

Problem: Find the derivative of $\tan(x)$.

Steps

1. Rewrite as a quotient
   $$\frac{d}{dx} \left( \frac{\sin(x)}{\cos(x)} \right)$$
2. Apply the quotient rule
   $$\frac{\cos(x) \cdot \frac{d}{dx}(\sin(x)) - \sin(x) \cdot \frac{d}{dx}(\cos(x))}{(\cos(x))^2}$$
3. Evaluate the derivatives of sine and cosine
   $$\frac{\cos(x) \cdot \cos(x) - \sin(x) \cdot (-\sin(x))}{\cos^2(x)}$$
4. Simplify the numerator
   $$\frac{\cos^2(x) + \sin^2(x)}{\cos^2(x)}$$
5. Apply the Pythagorean trigonometric identity
   $$\frac{1}{\cos^2(x)}$$
6. Rewrite using reciprocal trigonometric identities
   $$\sec^2(x)$$