The quotient rule of limit says that the limit of the quotient of two functions is the same as the quotient of the limit of the individual functions. In this post, we will prove the quotient law of limit by the epsilon-delta method. Question: What is an epsilon-delta proof of the division rule of limit? Here, … Read more
The product rule of limits says that the limit of the product of two functions is the same as the product of the limits of the individual functions. In this post, we will prove the product law of limits by the epsilon-delta method. Question: What is an epsilon-delta proof of the product law of limits? … Read more
The sum rule of limits says that the limit of the sum of two functions is the same as the sum of the limits of the individual functions. In this post, we will prove the sum/addition rule of limits by the epsilon-delta method. Question: What is an epsilon-delta proof of the sum law of … Read more
The cube root of x is a number whose cube is x, in other words, (cube root of x)^3 = x. In this post, we will evaluate the integration of the cube root of x. To do so, we will use the power rule of derivatives which is given below: $\int x^n dx = \dfrac{x^{n+1}}{n+1}+c$ … Read more
The derivative of root x minus 1 is 1 by 2 root x. In this post, we will prove this derivative formula by the power rule of derivatives as well as by the first principle of derivatives. Derivative of root(x)-1 Formula The Formula for the derivative of root x -1 is given below: Derivative of … Read more
The derivative of xsin x is sin x+x cos x. In this post, we will find the differentiation of x sin x using the first principle of derivatives. The derivative formula of xsin x is given below: $\dfrac{d}{dx}(x \sin x)$ $=\sin x+x\cos x$. We will now prove the above formula. Derivative of xsin x by … Read more
Note that n-th root can be written as raised to the power 1/n}. The derivative of n-th root of x is 1/n x^{1/n -1}. In this post, we will find the derivative of n-th root of x. Remark: The nth root of x is written as Derivative of nth root of x Using the rule … Read more
The derivative of fourth root of x is 1/(4 x^{3/4}). In this post, we will find the derivative of the fourth root of x using the power rule of derivatives. Note that fourth root of x can be written as Alternatively, one can write it as $x^{\dfrac{1}{4}}$. Derivative of Fourth Root of x To find … Read more
The derivative of x root x is 3 root(x)/2. In this post, we will find the derivative of x root(x) from first principle. This principle says that the derivative of f(x) is $\dfrac{d}{dx}(f(x))$ $=\lim\limits_{h \to 0}\dfrac{f(x+h)-f(x)}{h}$ $\quad \cdots (I)$ Derivative of x root x by First Principle Let $f(x)=x\sqrt{x}$ in the above formula (I). So … Read more
The derivative of e^4x with respect to x is 4e^4x. In this post, we will evaluate the derivative of e to the power 4x from first principle. Derivative of e^4x by first principle Let f(x) be a function in one variable x. The derivative of f(x) from first principle is given by the following … Read more