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)