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
The integration of tan^2x is equal to ∫tan2x dx = tan (x)-x. In this post, we will learn how to find the integral of tan square x. We will use the following two formulas: Integration of tan2x Question: What is the integration of tan2 x? That is, find ∫tan2 x dx. Answer: ∫tan2x dx = … Read more
The derivative of xn is equal to nxn-1. The function xn is read as x to the power n. The derivative of x to the n is referred to as the power rule of derivatives. In this post, we will find the derivative of xn by the limit definition of derivatives and the power rule. … Read more
In this blog post, we will find the derivative of sinxlogx, that is, sinx to the power logx, by the product rule of derivatives along with logarithmic differentiation. Let us recall the product rule of derivatives: The derivative of the product function f(x)g(x) is given as follows: $\dfrac{d}{dx}(f(x)g(x))$ $=f(x) g'(x) + f'(x) g(x)$ $\cdots (\star)$. … Read more
The derivative of x3/2 is equal to $\frac{3}{2} x^{1/2}$. In this blog post, we will find the derivative of x to the power 3/2 using the first principle and power rule. To find the derivative of $x^{\frac{3}{2}}$ using the limit definition, let us first recall the first principle of derivatives. The rule says that the … Read more
In this post, we will show that the trigonometric function sinx is continuous for all values of x. Here, we will use the limit method as well as the epsilon-delta definition. Note that $\lim\limits_{x \to c} \sin x=\sin c$ and $\sin x$ is defined for all real numbers. Thus, we can say that the function … Read more
In this post, we will learn about the epsilon-delta definition of continuity with solved examples. To learn this, let us first recall the definition of continuity. Definition of Continuity: A real-valued function f(x) is said to be continuous at a point x=a in the domain of f(x) if the following condition is satisfied: $\lim\limits_{x \to … Read more