Tutor profile: Sachchidanand P.

Prove that zero of a vector space is unique.

For proving the uniqueness in any context, we generally use the method of contradiction. So let us assume that there are two zeros, $$0_1$$ and $$0_2$$. We need to prove that $$0_1=0_2$$. For that observe, $( 0_1=0_1+0_2, \ \text{here we are using that } 0_2\ \text{is additive identity}. $)$(0_2=0_1+0_2,\ \text{here we are using that } 0_1\ \text{is additive identity}. $) Thus, $$0_1=0_2$$.

How to write a mathematical equation in Latex?

For writing a mathematical equation in Latex, there are a couple of ways that one can use. 1. Inline equation: For an inline equation, we use a single dollar sign. That is, suppose that we want to write "Let f(x)=sin(x) be a function." This can be written in latex as: Let $f(x)=\sin x$ be a function. 2. For the display mode, we use the double dollar. For example, $$ x^2+2x+3 .$$

Differentiate the function $$f(x)=\sin (x^2+2x+3)$$.

We want to find the derivative of the above function. So let us first analyze the given function. $(f(x)=\sin(g(x)),~ g(x)=x^2+2x+3$) We know the derivative of $$\sin x$$ which is $$\cos x$$. We also know the derivative of a polynomial $$g(x)=x^2+2x+3$$, which is $$2x+2$$. Here we have used the power rule and sum rule for the derivative. The first one states that the derivative of the function $$x^n$$ is $$nx^{n-1}$$ and the later one says that the derivative of the function $$\dfrac{d}{dx}(f(x)+g(x))=\dfrac{df(x)}{dx}+\dfrac{dg(x)}{dx}$$. Now we will apply the chain rule for finding out the derivative of the given function $$f(x)$$.The rule is given by $( \dfrac{d}{dx} a(b(x))=\dfrac{da}{dx}|_ {b(x)}\cdot \dfrac{db}{dx}$) If we apply the above rule then the drivative of $$f(x)$$ will be $( cos(g(x))\cdot \dfrac{dg(x)}{dx}=\cos(g(x))(2x+2). $) Hence the required derivative will be $$(2x+2)\cdot \cos(x^2+2x+3)$$.