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Chemistry
Braydon Henderson
1 July, 16:15
Derivative for tan (x)
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Noodles
1 July, 16:38
0
sec² (x)
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)
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