Answer

Verified

381.3k+ views

**Hint:**We explain the concept of derivation of a dependent variable with respect to an independent variable. We first find the formula for the derivation for ${{n}^{th}}$ power of a variable x where $\dfrac{d}{dx}\left( {{x}^{n}} \right)=n{{x}^{n-1}}$. We place the value for $n=-1,1$. We get the solution for the derivative of $f\left( x \right)=\dfrac{1}{x}$. We also explain the theorem with the help of the first order derivative.

**Complete step-by-step solution:**

Differentiation, the fundamental operations in calculus deals with the rate at which the dependent variable changes with respect to the independent variable. The measurement quantity of its rate of change is known as derivative or differential coefficients. We find the increment of those variables for small changes. We mathematically express it as $\dfrac{dy}{dx}$ where $y=f\left( x \right)$.

We need to find the derivative of $\left( \dfrac{1+3x}{3x} \right)\left( 3-x \right)$. We simplify the expression.

\[\left( \dfrac{1+3x}{3x} \right)\left( 3-x \right)=\dfrac{3-x}{3x}+\left( 3-x \right)=\dfrac{1}{x}-\dfrac{1}{3}+3-x\]. We need to find differentiation of constants which gives 0.

Let’s assume $y=f\left( x \right)=\dfrac{1}{x}-\dfrac{1}{3}+3-x$.

So, $\dfrac{dy}{dx}=\dfrac{d}{dx}\left( \dfrac{1}{x}-\dfrac{1}{3}+3-x \right)=\dfrac{d}{dx}\left( \dfrac{1}{x} \right)+\dfrac{d}{dx}\left( -\dfrac{1}{3}+3 \right)+\dfrac{d}{dx}\left( -x \right)$.

Differentiating we get $\dfrac{dy}{dx}=\dfrac{-1}{{{x}^{2}}}-1$.

**Therefore, the derivative of $\left( \dfrac{1+3x}{3x} \right)\left( 3-x \right)$ is $\dfrac{-1}{{{x}^{2}}}-1$.**

**Note:**If the ratio of $\dfrac{\Delta y}{\Delta x}$ tends to a definite finite limit when \[\Delta x \to 0\], then the limiting value obtained by this can also be found by first order derivatives. We can also apply first order derivative theorem to get the differentiated value.

We know that $\dfrac{dy}{dx}=\displaystyle \lim_{h\to 0}\dfrac{f\left( x+h \right)-f\left( x \right)}{h}$. Here $f\left( x \right)={{x}^{n}}$. Also, $f\left( x+h \right)={{\left( x+h \right)}^{n}}$. We assume $x+h=u$ which gives $f\left( u \right)={{\left( u \right)}^{n}}$ and $h=u-x$. As $h\to 0$ we get $u \to x$.

So, $\dfrac{df}{dx}=\displaystyle \lim_{h\to 0}\dfrac{f\left( x+h \right)-f\left( x \right)}{h}=\displaystyle \lim_{u \to x}\dfrac{{{u}^{n}}-{{x}^{n}}}{u-x}$.

We know the limit value $\displaystyle \lim_{x \to a}\dfrac{{{x}^{n}}-{{a}^{n}}}{x-a}=n{{a}^{n-1}}$.

Therefore, \[\dfrac{df}{dx}=\displaystyle \lim_{u \to x}\dfrac{{{u}^{n}}-{{x}^{n}}}{u-x}=n{{x}^{n-1}}\].

Recently Updated Pages

How many sigma and pi bonds are present in HCequiv class 11 chemistry CBSE

Why Are Noble Gases NonReactive class 11 chemistry CBSE

Let X and Y be the sets of all positive divisors of class 11 maths CBSE

Let x and y be 2 real numbers which satisfy the equations class 11 maths CBSE

Let x 4log 2sqrt 9k 1 + 7 and y dfrac132log 2sqrt5 class 11 maths CBSE

Let x22ax+b20 and x22bx+a20 be two equations Then the class 11 maths CBSE

Trending doubts

Which are the Top 10 Largest Countries of the World?

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

Write a letter to the principal requesting him to grant class 10 english CBSE

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

Difference Between Plant Cell and Animal Cell

Give 10 examples for herbs , shrubs , climbers , creepers

Change the following sentences into negative and interrogative class 10 english CBSE

How many crores make 10 million class 7 maths CBSE

Differentiate between homogeneous and heterogeneous class 12 chemistry CBSE