In this tutorial we are going to discuss about differentiating trigonometric identities. This online tutorial will help student to understand the differentiation of trigonometric function better and the solved problems and practice problems given here will prepare them for test. The rate at which the value of a real function changes with respect to its independent variable is called the derivative of the function. In this tutorial we shall define the derivative of a function in terms of a limit that involves the increments of the independent and the dependent variables.

how to find the derivatives of various types of functions.

If a variable x is changed from one value x0 to another value x1, then the difference x1 – x0 ( i.e. final value – initial value is called the increment of x and is denoted by deltax or h. thus, deltax = x_1 – x_0 or x_1 = x_0 + deltaxIf x increases from x_0 to x_1 then the increment is negative but if x decreases from x_0 to x_1 , then the increment is negative. In both the cases, we shall use the term increment for dx . Similarly, we shall denote the increment of y or f(x) by deltay or deltaf(x) . If in the value of a function y = f(x), the independent variable x is given an increment deltax or h from x = x_0 , then deltay or delta f (X) will denote the corresponding increment of y and we have.

y + deltay = f ( x_0 + deltax) where y = f(x_0)

deltay = f(x_0 + deltax) – y = f ( x_0 + deltax) – f(x_0)

f ‘(x) or (dy)/dx = lim_(deltax->0) ( f (x + deltax) – f(x))/(deltax) or lim_(h->0) (f(x+h) – f(x))/h

Derivatives of Trigonometric Functions Form:

d/dx ( cosx) = – sin x

d/dx ( sin x) = cos x

d/dx ( tan x)= sec2 x

d/dx ( cot x) = – cosec2 x

d/dx ( cosec x) = – cosec x * cot x

d/dx ( sec x) = sec x * tan x

Example Problems on Trigonometric Differentiation:

Pro 1: Differentiate : f (x) = x^2 cos x

Sol :

Let f ( x) = x^2 cos x Then f ( x+h) = ( x + h)2 cos ( x + h)

therefore f ‘ (x) = lim_(h->o) ( f ( x + h) – f(x))/ h = lim_(h->o) (( x+ h)^2 cos ( x + h) – x^2 cos x)/h

= lim_( h->0) (x^2 { cos (x + h) – cos x} + ( 2xh + h^2) cos ( x+ h))/h

= lim_(h->0) (x^2 * 2 sin ( x + h/2) * sin ( -h/2))/h + lim_(h->0) (h(2x + h) cos ( x + h) )/h

= -x^2 * lim_(h->0) sin (x + h/2) * lim_(h->0) (sin (h/2))/(h/2) + lim_(h->0) [(2x + h ) cos ( x + h)]

= -x^2 sin ( x+ 0 ) * lim_ ( theta->0) (sintheta)/theta + 9 2x + 0) cos ( x + 0) , where h/2 = theta ; as h-> 0 , theta -> 0

= – x^2 sinx * 1 + 2x cos x

= 2x cos x – x^2 sinx

Practice Problems on Trigonometric Differentiation:

Pro 1: Differentiate f (x) = tan ( 3x + 1) with resspect to x. ( Ans: 3 sec2 ( 3x + 1) )

Pro 2 : Differentiate f (x ) = sqrt( tanx) using the first principle of derivative with respect to x. ( Ans : (sec^2 x)/ (2( sqrt tanx)) )

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