promath The partial derivative of log(x^2+y^2) with respect to x is equal to 2x/(x2+y2) and with respect to y is equal to 2y/(x2+y2). So their formulas are as follows: Function Partial Derivative z=log(x2+y2) ∂z/∂x = 2x/(x2+y2) z=log(x2+y2) ∂z/∂y = 2y/(x2+y2) where ∂z/∂x is the partial derivative of z with respect to x. Partial Derivative of log(x2+y2) … Read more
The limit of (ex-1)/x, when x tends to zero, is equal to 1. That is, the formula of the limx→0 (ex-1)/x is given by limx→0 (ex-1)/x = 1. In this post, we will learn to find the limit of the function (ex-1)/x when x approaches to zero. Limit of (e^x-1)/x as x→0 Prove that limx→0 … Read more
For any x, sin(pi-x) formula is given below: sin(pi-x)=sinx Similarly, sin(pi-theta)=sin theta. In this post, we will learn how to compute sin(pi-x) or sin(pi-theta). sin(pi-x) Simplify To find the formula of sin(π-x), we will apply the following trigonometric identity: sin(A-B) = sinA cosB – cosA sinB …(∗) Here we put A=π and B=x; so that … Read more
The value of the limit of x/sinx as x approaches 0 is equal to 1. In this post, we will learn how to find the limit of $\frac{x}{\sin x}$ when x tends to 0. lim x/sinx when x approaches 0 Formula The formula of the limit of x/sinx when x tends to zero is given … Read more
The integral of e^(x^2), that is, the integration of e to the x2 is equal to $\dfrac{\sqrt{\pi}}{2}$ erfi(x)+C, where erfi(x) is called the “imaginary error function” and C is an integration constant. In this post, we will learn to integrate the function $e^{x^2}$. Integral of $e^{x^2}$ We know that ex can be written as ex … Read more
The integration of e2x is equal to e2x/2. In this post, we will learn how to integrate e to the 2x. The integration of e2x can be derived directly from the formula of the integral of emx which is given below: ∫emx dx = $\dfrac{e^{mx}}{m}$ +C where C is an integration constant. Putting m=2 in … Read more
The derivative of square root of ex is $\frac{1}{2}\sqrt{e^x}$. Here, we will find the derivative of root e to the power x by the chain rule of derivatives. What is the Derivative of root ex? We know that square root of x can be written as x1/2. So we have √ex = (ex)1/2 = ex/2. … Read more
The integration of the square root of a2+x2 is given as follows: $\int \sqrt{a^2+x^2} dx$ $=\dfrac{x}{2} \sqrt{a^2+x^2}$ $+\dfrac{a^2}{2}\log |x+\sqrt{a^2+x^2}|+C$, where $C$ is an integration constant. In this post, we will find the integral of root(a2+x2). Let’s learn how to integrate square root of a2+x2. Integration of $\sqrt{a^2+x^2}$ Question: What is the integration of square root … Read more
The derivative of cos2x is equal to -2sin2x. In this post, we will find the derivative of cos2x by the first principle, that is, by the limit definition of derivatives as well as by the chain rule of derivatives. Recall the first principle of derivatives. By this rule, we know that the derivative of a … Read more
The derivative of 3x is equal to 3x ln 3. Here ln 3 denotes the natural logarithm of 3, that is, ln 3 = loge 3. In this post, we will find the derivative of 3 to the power x by the first principle of derivatives. We first recall the first principle of derivatives. The … Read more