Differentiation of  Functions Using First Principles of Derivatives

To differentiate trigonometric functions using the first principles of derivatives, use the definition of the derivative: [Tex]f'(x) = \lim_{h \to 0} \frac{f(x+h) – f(x)}{h}[/Tex]

• Derivation of sin x: = cos x
• Derivative of cos x: = -sin x
• Derivative of tan x: = sec²x
• Derivative of cot x: = −cosec²x
• Derivative of sec x: = sec x.tan x
• Derivative of cosec x: = -cosec x.cot x

Derivative of Sinx by First Principle

Given: f(x) = sin(x)

Using the definition of the derivative:

[Tex]\frac{d}{dx}[/Tex] sin(x) = lim_{h → 0} [Tex]\frac{\sin(x + h) – \sin(x)}{h} [/Tex]

Applying the angle addition formula for sine:

lim_{h → 0} [Tex]\frac{\sin(x)\cos(h) + \cos(x)\sin(h) – \sin(x)}{h}[/Tex]

lim_{h → 0} [Tex]\frac{\sin(x)(\cos(h) – 1) + \cos(x)\sin(h)}{h}[/Tex]

Using the limits:

sin(x) lim_{h → 0} [Tex]\frac{\cos(h) – 1}{h}[/Tex] + cos(x) lim_{h → 0} [Tex]\frac{\sin(h)}{h} [/Tex]

As lim_{h → 0} [Tex]\frac{\sin(h)}{h}[/Tex] = 1 and lim_{h → 0} [Tex]\frac{\cos(h) – 1}{h}[/Tex] = 0 :

sin(x) · 0 + cos(x) · 1 = cos(x)

So, the derivative of sin(x) with respect to ( x ) using the first principles of derivatives is cos(x).

Derivative of Cosx by First Principle

[Tex]\frac{d}{dx}(\cos(x)[/Tex] = lim_{h → 0} [Tex]\frac{\cos(x + h) – \cos(x)}{h} [/Tex]

Using the angle addition formula for cosine:

= lim_{h → 0} [Tex]\frac{\cos(x)\cos(h) – \sin(x)\sin(h) – \cos(x)}{h}[/Tex]

= lim_{h → 0} [Tex]\frac{\cos(x)(\cos(h) – 1) – \sin(x)\sin(h)}{h}[/Tex]

= cos(x) lim_{h → 0} [Tex]\frac{\cos(h) – 1}{h}[/Tex] – sin(x) lim_{h → 0} [Tex]\frac{\sin(h)}{h}[/Tex]

As lim_{h → 0} [Tex]\frac{\sin(h)}{h}[/Tex] = 1 and lim_{h → 0} [Tex]\frac{\cos(h) – 1}{h}[/Tex] = 0 \):

= cos(x) ⋅ 0 – sin(x) ⋅ 1 = -sin(x)

So, the derivative of cos(x) with respect to ( x ) using the first principles of derivatives is -sin(x).

Derivative of Tanx by First Principle

[Tex] \frac{d}{dx}(\tan(x)) = \lim_{h \to 0} \frac{\tan(x + h) – \tan(x)}{h}[/Tex]

Using the tangent addition formula:

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

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

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

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

= [Tex]\frac{\sin(x)(1 – \cos(x))}{\cos^2(x)} + \frac{\cos(x)}{\cos^2(x)} \lim_{h \to 0} \frac{\sin(h)}{h}[/Tex]

= [Tex]\frac{\sin(x)(1 – \cos(x))}{\cos^2(x)} + \frac{\cos(x)}{\cos^2(x)} \cdot 1[/Tex]

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

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

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

= [Tex]\frac{\sin(x)}{\cos^2(x)} + \frac{\cos(x)}{\cos^2(x)} – \tan(x)\sec^2(x)[/Tex]

= [Tex]\frac{\sin(x) + \cos(x)}{\cos^2(x)} – \tan(x)\sec^2(x)[/Tex]

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

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

= [Tex]\frac{1}{\cos(x)}[/Tex]

= sec(x)

So, the derivative of tan(x) with respect to ( x ) using the first principles of derivatives is sec(x).

First Principle of Differentiation involves finding the derivative of a function using the fundamental definition of the derivative. This method requires calculating the limit of the difference quotient as the interval between two points on the function approaches zero.

In this article, we will learn about the first principle of derivative, its definition, its proof, how to find derivatives using the first principle, one-sided derivative and solved examples for better understanding.