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Question 511 Mark

Find the derivative of f(x) = $\frac{x+1}{x}$

Answer

The given function is defined everywhere except at x = 0. We use the quotient rule with u = x + 1 and v = x. Hence u′ = 1 and v′ = 1.
Therefore, we have,
$\frac{d f(x)}{d x}=\frac{d}{d x}\left(\frac{x+1}{x}\right)=\frac{d}{d x}\left(\frac{u}{v}\right)$ $=\frac{u^{\prime} v-u v^{\prime}}{v^{2}}=\frac{1(x)-(x+1) 1}{x^{2}}=-\frac{1}{x^{2}}$

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Question 521 Mark

Find the derivative of $f(x) = 1 + x + x^2 + x^3 + ... + x^{50} $ at $x = 1.$

Answer

Given, $f (a) = 1+ x + x^2 + x^3 +... + x^{50}$
On differentiating both sides w.r.t. $x$, we get
$f'(x) = 0 + 1 + 2x + 3x^2 +... + 50x^{49}$
At $x = 1,$
$f'(1) = 1 + 2(1)+ 3(1)^2 + ... + 50(1)^{49}$
$= 1 + 2 + 3 + ... + 50$
$= \frac { ( 50 ) ( 5 1) } { 2 } = 1275$$\left[ \because \sum _ { n } = \frac { n ( n + 1 ) } { 2 } \right]$

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Question 531 Mark

Compute the derivative of $6x^{100} – x^{55} + x.$

Answer

Using direct application of the theorem, the derivative of the function is $600 x^{99}-55 x^{54}+1$.

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Question 541 Mark

Find the limit: $\lim \limits_{x \rightarrow-1}\left[1+x+x^{2}+\ldots+x^{10}\right]$

Answer

$\lim \limits_{x \rightarrow-1}\left[1+x+x^{2}+\ldots+x^{10}\right]$ $=1+(-1)+(-1)^{2}+\ldots+(-1)^{10}$

= 1 - 1 + 1 - 1 + 1 - 1 + 1 - 1 + 1 - 1 + 1 = 1

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Question 551 Mark

Find the derivative of $f ( x ) = \frac { 1 } { x }$

Answer

We have, $f ( x ) = \frac { 1 } { x }$
By using the first principle,
$f ^ { \prime } ( x ) = \lim _ { h \rightarrow 0 } \frac { f ( x + h) - f ( x ) } { h }$
$\therefore \quad {f^\prime }(x) = \mathop {\lim }\limits_{h \to 0} \frac{{\frac{1}{{x + h}} - \frac{1}{x}}}{h}$ $\left[ {\begin{array}{*{20}{c}} {\because f(x) = \frac{1}{x}} \\ {\therefore f(x + h) = \frac{1}{{x + h}}} \end{array}} \right]$
$ = \mathop {\lim }\limits_{h \to 0} \frac{1}{h}\left[ {\frac{{x - (x + h)}}{{x(x + h)}}} \right] = \mathop {\lim }\limits_{h \to 0} \frac{1}{h}\left[ {\frac{{ - h}}{{x(x + h)}}} \right]$
$ = \mathop {\lim }\limits_{h \to 0} \left[ {\frac{{ - 1}}{{x(x + h)}}} \right] = \frac{{ - 1}}{{{x^2}}}$

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Question 561 Mark

Find the limit: $\lim \limits_{x \rightarrow 3}[x(x+1)]$

Answer

We have, $\lim \limits_{x \rightarrow 3}[x(x+1)]$ = 3 (3+1) = 3 (4) = 12

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Question 571 Mark

Find the derivative of the constant function f (x) = a for a fixed real number a.

Answer

f ′(x) = $\lim _{h \rightarrow 0} \frac{f(x+h)-f(x)}{h}$
$=\lim _{h \rightarrow 0} \frac{a-a}{h}=\lim _{h \rightarrow 0} \frac{0}{h}=0 \text { as } h \neq 0$

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Question 581 Mark

Find the limit: $\lim \limits_{x \rightarrow 1}\left[x^{3}-x^{2}+1\right]$

Answer

We have, $\lim \limits _{x \rightarrow 1}\left[x^{3}-x^{2}+1\right] = 1^3 – 1^2 + 1 = 1$

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Question 591 Mark

Find the derivative of $f(x) = x^2.$

Answer

$f ′(x) = \lim _{h \rightarrow 0} \frac{f(x+h)-f(x)}{h}$
$=\lim _{h \rightarrow 0} \frac{(x+h)^{2}-(x)^{2}}{h}=\lim _{h \rightarrow 0}(h+2 x)=2 x$

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MATHS 1 Marks Question