Answer:
Explanation:
Use the trigonometric ratio definition of the tangent function and the quotient rule.
Quotient rule: the derivative of a quotient is:
- [the denominator × the derivative of the numerator less the numerator × the derivative of the denominator] / [denominator]²
- (f/g)' = [ g×f' - f×g'] / g²
So,
- tan(x)' = [ sin(x) / cos(x)]'
- [ sin(x) / cos(x)]' = [ cos(x) sin(x)' - sin(x) cos(x)' ] / [cos(x)]²
= [ cos(x)cos(x) + sin(x) sin(x) ] / [ cos(x)]²
= [ cos²(x) + sin²(x) ] / cos²(x)
= 1 / cos² (x)
= sec² (x)
The result is that the derivative of tan(x) is sec² (x)
