Derivative of root x: The square root of x is a very important function in Mathematics. In this post, we will find the derivative of the square root of x using the first principle of derivatives and by the power rule of derivatives. At first, we find the derivative of root x by limit definition, … Read more
The derivative of e3x is 3e3x. The function e^3x is an exponential function with an exponent 3x. In this note, we will find the derivative of e to the power 3x by the first principle of derivatives and by the chain rule of derivatives. Derivative of e^3x using first principle As we know that the derivative … Read more
Note that the square root of 1+x can be written as (1+x)1/2. In this post, we will find the derivative of the square root of 1+x by the first principle of derivatives. Using this principle, the derivative of a function f(x) is $\dfrac{d}{dx}(f(x))$ $=\lim\limits_{h \to 0} \dfrac{f(x+h)-f(x)}{h}$ Derivative of sqrt(1+x) using Limit Definition Let $f(x)=\sqrt{1+x}$. … Read more
The function log(cos x) denotes the logarithm of the cosine function. Here we will find the derivative of log(cos x) using the first principle of derivatives. The derivative of a function $f(x)$ by the first principle of derivatives is defined to be the following limit: $f'(x)=\dfrac{d}{dx}(f(x))$ $=\lim\limits_{h \to 0} \dfrac{f(x+h)-f(x)}{h}$ $\quad \cdots (i)$ Here the symbol … Read more
If f(x) is a function of the real variable x, then its derivative by the first principle of the derivative is given by $f'(x)=\lim\limits_{h \to 0} \dfrac{f(x+h)-f(x)}{h}$ $\quad \cdots (i)$ Here $’$ denotes the derivative. In this post, we will find the derivative of \log(\sin x) by the first principle of derivatives. Derivative of log(sinx) … Read more
The derivative of square root of cosx is equal to (-sin x)/(2 root(cos x)).In this post, we will learn how to find the derivative of root cosx by first principle, that is, by the limit definition. The first principle of derivative is defined as follows: Let $f(x)$ be a differentiable function of $x$. The derivative … Read more
This page will discuss the continuity and differentiability of the absolute value of $x-a$, that is, of the function $|x-a|$, at the point $x=a$. Note that the function $|x-a|$ is defined as follows: $|x-a| = x-a$ if $x\geq a$ $=-(x-a)$ if $x<a$ Continuity of |x-a| at x=a Let us now discuss the continuity of the … Read more
The derivative of $\tan x$ is $\sec^2 x$, that is, $\dfrac{d}{dx}(\tan x)=\sec^2 x$. In this post, we will learn how to find the derivative of \tan x by the product rule of derivatives as well as the quotient and chain rule of derivatives. At first, we will recall the rules. Let $f(x)$ and $g(x)$ … Read more
The integration of root x is equal to 2/3 x3/2+C. In this post, we will calculate the integration of the square root of x. Note that the square root of x is an algebraic function. We will use the following power rule of derivatives: $\int x^n dx = \dfrac{x^{n+1}}{n+1}+c$ $ \cdots (i)$ where $c$ is … Read more
Here we will learn various types of calculus problems. For example, Limit (Link for Limits) Limit: definition, properties, and formulas Proofs of all limit formulas Solved Problems of Basic Limits Solved Problems of Trigonometric Limits Solved Problems of Exponential Limits Limits of Some Special Functions (to be updated) Intermediate Forms (to be updated) L’Hopital’s Rule with Applications (to … Read more