Derivative of \( \displaystyle \tan{\left(x \right)} \)

Problem 2.36 · easy

1. \[ \frac{d}{d x} \tan{\left(x \right)} \]
   derivativeStart with the derivative of the function.✓ Proved
2. \[ = \frac{d}{d x} \frac{\sin{\left(x \right)}}{\cos{\left(x \right)}} \]
   rewrite productRewrite tan(x) as sin(x)/cos(x). Rewrite the quotient as a product.✓ Proved
3. \[ = \sin{\left(x \right)} \frac{d}{d x} \frac{1}{\cos{\left(x \right)}} + \frac{\frac{d}{d x} \sin{\left(x \right)}}{\cos{\left(x \right)}} \]
   productApply the product rule.✓ Proved
4. \[ = \sin{\left(x \right)} \frac{d}{d x} \frac{1}{\cos{\left(x \right)}} + 1 \]
   derivative rewriteDifferentiate sin(x). Rewrite 1/cos(x) as cos(x)**(-1).✓ Proved
5. \[ = - \frac{\sin{\left(x \right)} \frac{d}{d x} \cos{\left(x \right)}}{\cos^{2}{\left(x \right)}} + 1 \]
   chainApply the chain rule to cos(x)**(-1).✓ Proved
6. \[ = \frac{\sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + 1 \]
   derivative algebra algebra algebra algebraDifferentiate cos(x). Simplify the signs and terms. Rearrange the terms. Rewrite the negative exponent. Combine the fraction.✓ Proved
7. \[ = \tan^{2}{\left(x \right)} + 1 \]
   rewriteRewrite the fraction as tan(x).✓ Proved
8. \[ = \sec^{2}{\left(x \right)} \]
   simplifyUse the trigonometric identity 1 + tan(x)**2 = sec(x)**2.✓ Proved