What is the first derivative of 3secx?

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Given the function

y = 3 · sec(x)

         dy
find  —— .
         dx

Using the definition of derivative:



Secant is the reciprocal of cosine:



Reduce those fractions in brackets to the same denominator:



Recall one trogonometric identity (cosine of a sum):

•  cos(x + h) = cos(x) · cos(h) – sin(x) · sin(h)

then you have



Take out the common factor  cos(x)  from the first two terms in that numerator:



Now, split up that first fraction into a sum of two fractions:



Multiply and divide that first term in that sum by



But  1 – cos²(h) = sin²(h):



                                                    sin(h)
Take out the common factor  ————  :
                                                       h



The limit of  sin(h)/h  as  h  approaches  0  equals  1  (fundamental trigonometric limit). So the derivative is



∴                ✔

I hope this helps. =)

Tags:  derivative function definition limit secant trigonometric trig differential integral calculus