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
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