Differentiation Of Trigonometric Functions

Answer

Let \( y = \sin x \). For small \( \delta x \), \( y + \delta y = \sin(x + \delta x) \) \( \delta y = \sin(x + \delta x) - \sin x \) Using identity: \( \sin C - \sin D = 2\cos\frac{C+D}{2}\sin\frac…

Answer

Let \( y = \cos x \). Then \( y + \delta y = \cos(x + \delta x) \) \( \delta y = \cos(x + \delta x) - \cos x \) Using identity: \( \cos C - \cos D = -2\sin\frac{C+D}{2}\sin\frac{C-D}{2} \) \( \delt…

Answer

Let \( y = \tan x \). Then \( y + \delta y = \tan(x + \delta x) \) \( \delta y = \tan(x + \delta x) - \tan x = \frac{\sin(x + \delta x)}{\cos(x + \delta x)} - \frac{\sin x}{\cos x} \) \( = \frac{\si…

Answer

Using chain rule: Let \( u = 2x \Rightarrow \frac{du}{dx} = 2 \) \( y = \tan u \Rightarrow \frac{dy}{du} = \sec^2 u \) \( \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} = \sec^2(2x) \cdot 2 = 2\…

Answer

Let \( y = \sec 3x \Rightarrow y + \delta y = \sec 3(x + \delta x) \) \( \delta y = \sec(3x + 3\delta x) - \sec 3x = \frac{1}{\cos(3x + 3\delta x)} - \frac{1}{\cos 3x} \) \( = \frac{\cos 3x - \cos(3…

Answer

Using product rule: Let \( u = x^4 \), \( v = \sin 2x \) \( \frac{du}{dx} = 4x^3 \), \( \frac{dv}{dx} = 2\cos 2x \) \( \frac{dy}{dx} = u \frac{dv}{dx} + v \frac{du}{dx} = x^4(2\cos 2x) + \sin 2x(4x…

Answer

Simplify first: \( \frac{\sin x}{1 + \cos x} = \frac{2\sin\frac{x}{2}\cos\frac{x}{2}}{2\cos^2\frac{x}{2}} = \tan\frac{x}{2} \) Now differentiate: \( \frac{dy}{dx} = \sec^2\frac{x}{2} \cdot \frac{1}…

Answer

Using quotient rule: \( \frac{dy}{dx} = \frac{x \cdot \cos x - \sin x \cdot 1}{x^2} = \frac{x\cos x - \sin x}{x^2} \) Answer: \( \frac{dy}{dx} = \frac{x\cos x - \sin x}{x^2} \)…