promath 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 formula of cos(a+b)cos(a-b) is given by cos(a+b)cos(a-b) = cos2a -sin2b. In this post, we will establish the formula of cos(a+b) cos(a-b). Note that cos(a+b) cos(a-b) is a product of two cosine functions. We will use the following two formulas: cos(a+b) = cos a cos b – sin a sin b …(i) cos(a-b) = cos a … Read more
In this post, we will establish the formula of sin(a+b) sin(a-b). Note that sin(a+b) sin(a-b) is a product of two sine functions. We will use the following two formulas: sin(a+b) = sin a cos b + cos a sin b …(i) sin(a-b) = sin a cos b – cos a sin b …(ii) Formula of sin(a+b) … 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
Answer: The derivative of e^2x by first principle is equal to 2e2x. In this article, we will find the derivative of e to the power 2x using the first principle. Recall that for a function f(x) in one variable x, the derivative of f(x) from first principle is given by the limit below: $\dfrac{d}{dx}(f(x))$ $=\lim\limits_{h … Read more