Question

Derivative for tan(x)

Answer 1

The derivative of tan(x) is sec²(x), which is derived by using the quotient rule on the trigonometric identity tan(x) = sin(x)/cos(x).

Step-by-step explanation:

The derivative of tan(x) with respect to x can be found by using the quotient rule since tan(x) = sin(x)/cos(x). Applying the quotient rule, we get:

• The derivative of the numerator (sin(x)) with respect to x is cos(x).
• The derivative of the denominator (cos(x)) with respect to x is -sin(x).
• Then the derivative of tan(x) is [cos(x) × cos(x) + sin(x) × sin(x)] / (cos(x))².
• This simplifies to 1/(cos(x))² or sec²(x).

Therefore, the derivative of tan(x) is sec²(x).

Answer 2

• sec² (x)

Step-by-step explanation:

Use the trigonometric ratio definition of the tangent function and the quotient rule.

• tan(x) = sin(x) / cos(x)

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)