Differentiation of Common Functions

Differentiation of some common functions is listed in the following table:

Other then above mentioned functions, there are more function which can be diferntiated in differential calculus. Some of these functions are:

• Logarithmic Functions
• Exponential Functions
• Polynomial Functions

Differentiation of Exponential Function

Differentiation of exponential function is depends on base of the function.

• If base is Euler number (i.e. e = 2.71828), them the derivative of e^x is same e^x .
• If base is not Euler number ( say a),, then the derivative of function a^x is given by a^x ln a where ln a is natural log of a i.e. base is e.

Differentiation of Logarithmic Function

Differentiation of Logarithmic function is depends the base of logarithm same as exponential function.

• If base is Euler constant ( e=2.71828), the the derivative is give by [Tex]\frac{d}{dx}(\ln{x})=\frac{1}{x} [/Tex]
• If base is other than Euler number (i.e. base is not e) say(a), then derivative can be written as [Tex]\frac{d}{dx}(\log_{a}x)=\frac{1}{x\ln{a}}[/Tex], where ln(a) is natural log i.e. base e.

Differentiation of Polynomial Function

Differentiation of Polynomial function can be evaluated by using Power rule of Derivative for all the terms.

Polynomial of order n can be written as

P(x)=a_{0}+a_{1}x+a_{2}x^2+\cdots+a_{n}x^n

Derivative of P(x) can be written as:

[Tex]P'(x)=\frac{d}{dx}{a_{0}+a_{1}x+a_{2}x^2+\cdots+a_{n}x^n}=0+1a_{1}+a_{2}2x+\cdots+a_{n}nx^{n-1}[/Tex]

⇒ [Tex]P'(x)=1a_{1}+2a_{2}x+\cdots+na_{n}x^{n-1}[/Tex]

Differential calculus is a branch of calculus that studies the concept of a derivative and its applications. Derivative tells us about the rate at which a function changes at any given point. Differential Calculus is crucial to many scientific and engineering areas since it allows for the estimation of instantaneous rates of change and curve slopes. In this article, we will be discussing about Differential Calculus and its fundamentals, which every students should know.

This article helps learners in understanding of differential calculus, its concepts, and its applications. By the end of this article, readers should be able to understand the fundamentals of derivatives and use them to solve real-world issues.