# Derivatives of Trigonometric Functions

Sub Topics

The differentiation of the trigonometric functions is called as finding the Derivatives Of Trigonometric Functions. The derivative of a function shows how does a function is changed when there occurs a change in one of its variables. As y = f(x) is a function and x is changing by x so there will a change in y by y. Such as sinx, cosx, tanx, etc. Following are the differential coefficients of the basic trigonometric functions which are used as formulas in the differential calculus.

## Derivative of Sinx

The study of Derivative of Sinx is a type of Trigonometric Function studies under Derivatives Of Trigonometric Functions:

Let f(x) = sinx

So, f(x+h) = sin(x+h)

Now, by the first principle of derivative

$\frac{d}{dx}$f(x) = $\mathop{\lim}\limits_{h \to 0}$.$\frac{f(x+h)?f(x)}{h}$

= $\mathop{\lim}\limits_{h \to 0}$.$\frac{sin(x+h)?sinx}{h}$

= $\mathop{\lim}\limits_{h \to 0}$.$\frac{2cos(x+\frac{h}{2})sin\frac{h}{2}}{h}$

= $\mathop{\lim}\limits_{h \to 0}$.cos(x+$\frac{h}{2}$).$\mathop{\lim}\limits_{h \to 0}$.$\frac{sin\frac{h}{2}}{\frac{h}{2}}$

= cos(x+$\frac{0}{2}$).1, [since, $\mathop{\lim}\limits_{h \to 0}$.$\frac{sin\frac{h}{2}}{\frac{h}{2}}$ = 1]

= cosx,

So we have,

$\frac{d}{dx}$ (sinx) = cosx

## Derivative of Cosx

The study of Derivative of Cosx is a type of Trigonometric Function studies under Derivatives Of Trigonometric Functions:

Let f(x) = cosx

So, f(x+h) = cos(x+h)

Now, by the first principle of derivative

$\frac{d}{dx}$f(x) = $\mathop{\lim}\limits_{h \to 0}$.$\frac{f(x+h)?f(x)}{h}$

= $\mathop{\lim}\limits_{h \to 0}$.$\frac{cos(x+h)?cosx}{h}$

= $\mathop{\lim}\limits_{h \to 0}$.$\frac{2sin(x+\frac{h}{2})sin(\frac{?h}{2})}{h}$

= - $\mathop{\lim}\limits_{h \to 0}$.sin(x+$\frac{h}{2}$).$\mathop{\lim}\limits_{h \to 0}$.$\frac{sin\frac{h}{2}}{\frac{h}{2}}$

= - sin(x+$\frac{0}{2}$).1, [since, $\mathop{\lim}\limits_{h \to 0}$.$\frac{sin\frac{h}{2}}{\frac{h}{2}}$ = 1]

= - sinx

So

$\frac{d}{dx}$(cosx) = - sinx

By using the first principle of derivative, we can find the derivatives of trigonometric functions. The derivatives of the trigonometric functions are as following which are used in differential calculus.

1. $\frac{d}{dx}$(sinx) = cosx

2. $\frac{d}{dx}$(cosx) = - sinx

3. $\frac{d}{dx}$(tanx) = sec2x

4. $\frac{d}{dx}$(cosecx) = - cosecx.secx

5. $\frac{d}{dx}$(secx) = secx.tanx

6. $\frac{d}{dx}$(cotx) = - cosec2x