Derivative of Root(x+1) by First Principle

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

Derivative of log(cos x) by First Principle

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

Derivative of log(sin x) by First Principle

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

Derivative of 1/sqrt(1-x^2) | Derivative of 1/sqrt(a^2-x^2)

Derivative of  $1/\sqrt{1-x^2}$ First Method: At first, we will find the derivative of 1/root(1-$x^2$) by the quotient rule of derivatives. Let us recall the quotient rule of derivatives. If $f$ and $g$ be two functions then the derivative of $f/g$ is given by the following formula: $\dfrac{d}{dx}(f/g)$ $=\dfrac{gf’ -f g’}{g^2}$ $\cdots (\star)$ Here $’$ denotes the … Read more

Derivative of square root of x^2+y^2 | Derivative of x/sqrt{x^2+y^2}

In this post, we will learn how to find the derivative of various square roots; for example, the derivatives of root(x^2+y^2), x/root(x^2+y^2) and y/root(x^2+y^2).   For more details of square roots, please visit the page Square Root of x: Definition, Symbol, Graph, Properties, Derivative, Integration. Derivative of $\sqrt{x^2+y^2}$ To find the derivative of $\sqrt{x^2+y^2}$ with respect … Read more

Derivative of root(x)+1/root(x)

Here, we will find the derivative of root x+1/root x with respect to x. In the end, we will also evaluate this derivative at x=1. For more details of square roots, please click on the page Square Root of x: Definition, Symbol, Graph, Properties, Derivative, Integration Root(x)+1/Root(x) Derivative Question: Find the derivative of $\sqrt{x}+\frac{1}{\sqrt{x}}$ Answer: At … Read more

Derivative of square root of sin x using first principle

In this section, we will learn how to find To answer the question, let us first know the definition of the derivative. Definition of derivative: Let $f(x)$ be a differentiable function of $x$. From first principle of by definition, the derivative of $f(x)$ is given as follows: $f'(x)=\lim\limits_{h \to 0} \dfrac{f(x+h)-f(x)}{h}$ Derivative of the Square … Read more

Modulus of x is not Differentiable Proof

In this section, we will prove that the absolute value of x is not differentiable at the point x=0. In other words, the function |x| is not differentiable at x=0. Absolute Value of x is not Differentiable at 0 The function f(x)=|x| is defined as follows: $|x|=\begin{cases} x, & \text{ if } x\geq 0 \\ … Read more