promath The simplification of 1+tan2 x is equal to sec2 x. In this post, we will find the formula of 1+tan2x. This will be proved using trigonometric identities. 1+tan2x Formula To simplify the expression 1+tan2 x, we will follow the below steps: Step 1: At first, we put $\tan x=\dfrac{\sin x}{\cos x}$. By doing so, we … Read more
The value of cos(x-pi) is equal to -cosx. This can be done using the trigonometric formulas of compound angles. In this post, we will learn how to simplify cos(pi-x) and cos(x-pi). We will use the following formula on compound angles to find cos(pi-x) and cos(x-pi). $\cos(a-b)=\cos a \cos b +\sin a \sin b$ $\cdots (\star)$ … 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
The derivative of cos(ex) is equal to -ex sin(ex). In this post, we will learn how to find the derivative of cos(ex) by the chain rule of derivatives. Derivative of cos(ex) Question: Find the derivative of cos(ex). Answer: The derivative of cos(ex) is equal to -exsin(ex). Explanation: Note that f(x)=cos(ex) is a composite function. The … Read more
The derivative of e1/x is $-\frac{1}{x^2} e^{1/x}$. So if y=e1/x, then dy/dx = -1/x2 e1/x. In this post, we will learn how to differentiate e to the power 1/x. Derivative of e1/x Question: What is the derivative of e raised to 1/x? Answer: The derivative of e raised to 1/x is equal to $-1/x^2 e^{1/x}$. … Read more
The derivative of pi/4 is equal to zero. This is because pi/4 is a constant. In this post, we will learn how to find the derivative of pi divided by 4. Derivative of pi/4 Answer: The derivative of pi/4 is 0. Explanation: We know that the value of the number $\pi$ is approximately … Read more
The derivative of cos(x^4) is equal to -4x^3 sin(x^4). In this post, we will find the derivative of cosx^4 by the chain rule. Derivative of cos(x^4) To find the derivative of cos(x^4), we will use the chain rule of derivatives. The chain rule is used to find the derivative of a composite function which we … Read more