Here we will calculate the derivatives of some well-known functions from the first principle. For example, we will find the derivatives of the polynomial functions, trigonometric functions, exponential functions, logarithmic functions, and so on. Firstly, we find the derivative of xn using the definition of the derivative. Power rule of Derivative using First Principle: \[\frac{d}{dx}(x^n)=nx^{n-1}\] … Read more
Here we will prove various properties of derivatives with applications one by one. Derivative of a constant function is zero- proof: For a constant $c$, we have $\frac{d}{dx}(c)=0$ Proof: Let $f(x)=c$ Now, $\frac{d}{dx}(c)$ $=\frac{d}{dx}(f(x))$ $=\lim\limits_{h \to 0}{ \large \frac{f(x+h)-f(x)}{h} }$ $=\lim\limits_{h \to 0}{ \large \frac{c-c}{h} }$ $=\lim\limits_{h \to 0}{ \large \frac{0}{h} }$ $=\lim\limits_{h \to 0}0$ … Read more
The concept of the derivative is the backbone of the theory of Calculus. Here we will learn the definition of the derivative of a function, its various formulas, and its properties with examples. Definition of the Derivative of a function: Limit definition of derivative: Let f(x) be a function of the variable x. Then … Read more
The theory of Differentiation is the backbone of Calculus. With the help of differentiation, we actually determine the rate of changes of the dependent variable with respect to the independent variable. In this section, we will discuss the concept of derivatives. Here we go. 👩 Few Definitions: The increment of a variable: Let $x$ be a … Read more