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
In this post, we will find the integral of root(a2-x2). The integration of square root of a2-x2 is given as follows ∫ √(a2-x2) dx = x/2 ⋅ √(a2-x2) + a2/2 sin-1(x/a) + C, where C is an integration constant. Integration of $\sqrt{a^2-x^2}$ Let $I = \int \sqrt{a^2-x^2} dx$ $\cdots (I)$ We will find the integration … Read more
The derivative of 2x is equal to 2xln2 where ln 2 is the natural logarithm of 2, that is, ln 2 = loge2. In this post, we will find the derivative of 2x by the first principle of derivatives. Derivative of 2x from First Principle We know that the derivative of a function f(x) by … Read more
The derivative of xex is equal to (1+x)ex. In this post, we will find the derivative of xex by the first principle and by the product rule of derivatives. The first principle of derivatives says that the derivative of a function f(x) is given by the following limit formula: $\dfrac{d}{dx}(f(x))$ $=\lim\limits_{h \to 0} \dfrac{f(x+h)-f(x)}{h}$ … Read more
The integration of 1/(1+x2) is equal to tan2 x. In this post, we will see how to integrate 1/(1+x^2). Integration of $\dfrac{1}{1+x^2}$ Question: What is the integration of $\dfrac{1}{1+x^2}$? That is, Find $\int \dfrac{1}{1+x^2} dx$ Answer: The integration of $\dfrac{1}{1+x^2}$ is $\tan^2 x$. Explanation: Let us substitute $x=\tan t$ $\cdots (\star)$ Differentiating with respect to … Read more
The derivative of 10x is equal to 10x ln 10. Here ln denotes the logarithm with base e, called the natural logarithm. In this post, we will learn to compute the derivative of 10 raised to x. Derivative of 10x Question: What is the derivative of 10x? Answer: The derivative of 10x is 10x ln … Read more
The Maclaurin series expansion of sinx or the Taylor series expansion of sinx at x=0 is given as follows: $\sin x= \sum_{n=0}^\infty \dfrac{(-1)^n}{(2n+1)!}x^{2n+1}$ $=x-\dfrac{x^3}{3!}+\dfrac{x^5}{5!}-\cdots$ Taylor Series Expansion of Sinx at x=0 We know that the Maclaurin series expansion of $\sin x$ or the Taylor series of a function $f(x)$ at $x=0$ is given by the … Read more
The integration of e3x is e3x/3. In this post, we will learn how to find the integral of e to the 3x. Let us recall the formula of the integral of emx: $\int e^{mx} dx=\dfrac{e^{mx}}{m}+C$ where C is an integral constant. Thus, the integral of e3x will be equal to $\int e^{3x} dx=\dfrac{e^{3x}}{3}+C$. What is the … Read more
The derivative of 2x is equal to 2x ln2. Here, ln denotes the natural logarithm (logarithm with base e). In this post, we will find the derivative of 2 to the power x. Derivative of 2x Formula As we know that the derivative of ax is axln a, the formula for the derivative of 2x … Read more
The derivative of sin5x is equal to 5cos5x. In this post, we will find the derivative of sin5x by the first principle, that is, by the limit definition of derivatives. By the first principle of derivatives, we know that the derivative of a function f(x) is given by the following limit: $\dfrac{d}{dx}(f(x))$$=\lim\limits_{h\to 0} \dfrac{f(x+h)-f(x)}{h}$ …(I) Derivative … Read more