# Derivative Trig

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Derivative trig is used in finding derivative of trigonometric functions. The derivative of trig functions can be found by using definition of derivative and by using limit rules. It is a process of finding rate of change of trigonometric function with respect to a variable. Trigonometry is useful in areas such as astronomy, surviving, physics etc.  Remember derivatives of basic 6 trigonometric functions. One derivative is shown in example 1.

Problem 1: Find the derivative of d (sin x) / dx

Solution: Given: d (sin x) / dx.

=> By the definition of derivative, d f(x) / dx = lim h->0 f(x + h) – f(x) / h

=> d (sin x) / dx = lim h->0 (sin (x+ h) – sin x) / h

= lim h->0 (sin x cos h + sin h cos x – sin x) / h   (using trigonometric identity)

= lim h->0 (sin x (cos h – 1) + sin h cos x) / h

= sin x lim h->0 (cos h – 1) / h + cos x lim h->0 (sin h) / h   (By separating the limits

=> By applying trigonometric limits we get, d (sin x) / dx = sin x. 0 + cos x. 1 = cos x

Problem 2: Find the derivative of d (2sec(x) – 5 cot (x))/dx

Solution: Given: d (2sec(x) – 5 cot(x))/dx

=> We know the derivative of basic 6 trigonometric functions

=> So, d sec (x) / dx = sec(x) tan(x) and also d cot(x)/dx = - csc^2(x)

=> d (2sec(x) – 5 cot(x))/dx = 2 sec(x) tan(x) – 5(- csc^2(x))

= 2 sec(x) tan(x) + 5 csc^2(x)

=> Therefore, the derivative of d (2sec(x) – 5 cot (x))/dx = 2 sec(x) tan(x) + 5 csc^2(x)
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