Differentiate x^2-1 x from first principle. - Vidyadip

Question

Differentiate $\frac{x^2-1}{x}$ from first principle.

✓

Answer

We need to find derivative of $f(x)=\frac{x^2-1}{x}$
Derivative of a function f(x) from first principle is given by
$f ^{ f }( x )=\underset{{h \rightarrow 0}}{\lim}=\frac{f(x+h)-f(x)}{h}$ \{where h is a very small positive number $\}$
$\therefore$ Derivative of $f(x)=\frac{x^2-1}{x}$ is given as
$f ^{\prime}( x )=\underset{{h \rightarrow 0}}{\lim}=\frac{f(x+h)-f(x)^x}{h}$
$\begin{array}{l}\Rightarrow f ^{\prime}( x )=\underset{{h \rightarrow 0}}{\lim}\frac{\frac{(x+h)^2-1}{x+h}-\frac{x^2-1}{\alpha}}{h} \\
\Rightarrow f ^{\prime}(x)=\underset{{h \rightarrow 0}}{\lim}\frac{\left\{(x+h)^2-1\right\} x-(x+h)\left(x^2-1\right)}{h x(x+h)} \\
\Rightarrow f ^{\prime}(x)=\underset{{h \rightarrow 0}}{\lim} \frac{\left\{(x+h)^2-1\right\} x-(x+h)\left(x^2-1\right)}{h} \times \underset{{h \rightarrow 0}}{\lim} \frac{1}{x(x+h)} \\
\Rightarrow f ^{\prime}(x)=\frac{1}{x^2} \underset{{h \rightarrow 0}}{\lim} \frac{\left\{(x+h)^2-1\right\} x-(x+h)\left(x^2-1\right)}{h} \\
\Rightarrow f^{\prime}(x)=\frac{1}{x^2} \underset{{h \rightarrow 0}}{\lim} \frac{\left\{x^2+h^2+2 x h-1\right\} x-\left\{x^3+h x^2-x-h\right\}}{h} \\
\Rightarrow f^{\prime}(x)=\frac{1}{x^2} \underset{{h \rightarrow 0}}{\lim} \frac{x^3+h^2 x+2 x^2 h-x-x^3-h x^2+x+h}{h} \\
\Rightarrow f ^{\prime}(x)=\frac{1}{x^2} \underset{{h \rightarrow 0}}{\lim} \frac{h^2 x+x^2 h+h}{h} \\
\Rightarrow f ^{\prime}(x)=\frac{1}{x^2} \underset{{h \rightarrow 0}}{\lim} \frac{h\left(h x+x^2+1\right)}{h} \\
\Rightarrow f ^{\prime}( x )=\frac{1}{x^2} \underset{{h \rightarrow 0}}{\lim}\left(h h+x^2+1\right) \\
\Rightarrow f^{\prime}(x)=\frac{1}{x^2}\left(0 \times x+x^2+1\right)=\frac{x^2+1}{x^2}=1+\frac{1}{x^2}\end{array}$
$\therefore f^{\prime}(x)=1+\frac{1}{x^2}$
Hence,
Derivative of $f(x)=\frac{x^2-1}{x^2}=1+\frac{1}{x^2}$

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "3 Marks Question" from Model Paper 2→Model Paper 2 — all question types→MATHS STD 11 Science question papers→Rajasthan Board STD 11 Science question papers→Rajasthan Board question paper generator→

Similar questions

Evaluate the following limit:
$\lim\limits_{\text{x}\rightarrow3}\frac{\text{x}^2-4\text{x}+3}{{\text{x}^2}-2\text{x}-3}$

Find the term independent of x in the expansion of the following expressions:

$\Big(3\text{x}-\frac{2}{\text{x}^{2}}\Big)^{15}$

Show that for any sets A and B, A = ( A $\cap$ B ) $\cup$ ( A – B ) and A $\cup$ ( B – A ) = ( A $\cup$ B )

The common ratio of a G.P. is 3 and the last term is 486. If the sum of these terms be 728, Find the first term.

Find the coefficient of x¹⁵ in the expansion of (x - x²)¹⁰.

Sketch the graphs of the following function:
$\text{f(x)}=2\sin\pi\text{x},0\leq\text{x}\leq2$

Find the sum of all two digit numbers which when divided by 4, yield 1 as remainder.

Find the equation of the circle passing through the points (4, 1) and (6, 5) and whose centre is on the line 4x + y = 16.

Show the following quadratic equation by factorization method:
21x² - 28x + 10 = 0

Let f, g: $R \to R$ be defined, respectively by f(x) = x + 1, g(x) = 2x – 3. Find f + g, f – g and $\frac{f}{g}$.