17. Derivative of Trigonometric Functions

Dated: 30-10-2024

Derivative of \(f(x) = \sin(x)\)

\[\frac{d}{dx} (\sin x) = \lim_{h \to 0} \frac{\sin(x+h) - \sin(x)}{h}\]

\[ = \lim_{h \to 0} \frac{\sin(x)\cos(h) + \cos(x)\sin(h) - \sin(x)}{h}\]

\[ = \lim_{h \to 0} \frac{\sin(x)\cos(h) - \sin(x) + \cos(x)\sin(h)}{h}\]

\[ = \lim_{h \to 0} \left[\sin(x)\left(\frac{\cos(h)-1}{h}\right) + \cos(x)\left(\frac{\sin(h)}{h}\right)\right]\]

\[ = \lim_{h \to 0} \left[\cos(x)\left(\frac{\sin(h)}{h}\right) - \sin(x)\left(\frac{1-\cos(h)}{h}\right)\right]\]

\[\because \lim_{h \rightarrow 0} \cos(x) = \cos(x)\]

\[\because \lim_{h \rightarrow 0} \sin(x) = \sin(x)\]

And both are constants,

\[ = \cos(x)\lim_{h\to0}\left(\frac{\sin (h)}{h}\right)-\sin(x)\lim_{h\to0}\left(\frac{1-\cos (h)}{h}\right)\]

\[= \cos(x) \cdot (1) - \sin(x) \cdot (0)\]

\[= \cos(x)\]

Derivative of \(f(x) = \cos(x)\)

\[\frac{d}{dx}\cos(x) = \lim_{h\to0}\frac{\cos(x+h)-\cos(x)}{h}\]

\[= \lim_{h\to0}\frac{\cos(x)\cos(h)-\sin(x)\sin(h)-\cos(x)}{h}\]

\[= \lim_{h\to0}\left(\frac{\cos(x)\cos(h)-\cos(x)}{h}-\frac{\sin(x)\sin(h)}{h}\right)\]

\[= \lim_{h\to0}\left(\cos(x)\left(\frac{\cos(h)-1}{h}\right)-\frac{\sin(x)\sin(h)}{h}\right)\]

\[= \cos(x)(0) - \sin(x)(1)\]

\[= -\sin(x)\]

Derivative of \(f(x) = \tan(x)\)

\[\frac{d}{dx}\tan(x) = \frac{d}{dx}\left[\frac{\sin(x)}{\cos(x)}\right]\]

using the quotient formula¹

\[= \frac{\cos(x)\frac{d}{dx}\sin(x)-\sin(x)\frac{d}{dx}\cos(x)}{\cos^{2}(x)}\]

\[= \frac{\cos(x)\cos(x)-\sin(x)[-\sin(x)]}{\cos^{2}(x)}\]

\[= \frac{\cos^{2}(x)+\sin^{2}(x)}{\cos^{2}(x)}\]

\[\because \cos^{2}(x)+\sin^{2}(x) = 1\]

\[=\frac{1}{\cos^{2}(x)}\]

\[= \sec^2(x)\]

Derivative of \(f(x) = \sec(x)\)

\[\frac{d}{dx}\sec(x) = \frac{d}{dx}\left(\frac{1}{\cos(x)}\right)\]

Using the quotient formula ¹

\[=\frac{cos(x)(0)-(1)[-sin(x)]}{\cos^{2}(x)}\]

\[=\frac{\sin(x)}{\cos(x)}\cdot\frac{1}{\cos(x)}\]

\[=\sec(x)\tan(x)\]

Derivative of \(f(x) = \csc(x)\)

\[\frac{d}{dx}\csc(x) = \frac{d}{dx}\left(\frac{1}{\sin(x)}\right)\]

\[=\frac{\sin(x)(0)-(1)[\cos(x)]}{\sin^{2}(x)}\]

\[=-\frac{\cos(x)}{\sin(x)}\cdot\frac{1}{\sin(x)}\]

\[=-\csc(x)\cot(x)\]

Derivative of \(f(x) = \cot(x)\)

\[\frac{d}{dx}\cot(x) = \frac{d}{dx}\left(\frac{1}{\tan(x)}\right)\]

\[=\frac{\tan(x)(0)-(1)[\sec^{2}(x)]}{\tan^{2}(x)}\]

\[=-\frac{\sec^{2}(x)}{\tan^{2}(x)}\]

\[= - \frac{1}{\cos^2(x)} \div \frac{\sin^2(x)}{\cos^2(x)}\]

\[= - \frac{1}{\cos^2(x)} \times \frac{\cos^2(x)}{\sin^2(x)}\]

\[= - \frac{1}{\sin^2(x)}\]

\[=-\csc^{2}(x)\]