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
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
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
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
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
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
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
Derivative implies Continuity but Converse NOT True: In this section, we will prove that if a function is differentiable at a point, then the function is continuous at that point. Its converse statement is the following: if a function is continuous, then it is not necessarily differentiable. We prove the converse statement by providing examples. … Read more