Question

find differentiation with respect to x.(d/dx)​

Answer 1

Explanation:

$\bold{1. \: {x}^{ \frac{7}{2} } }$

$\large\pink {\boxed{ \bold{ \green{Solution : }}}}$

$\bold{Suppose \: y = {x}^{ \frac{7}{2} } }$

Differential with respect to x,

$\bold{ \frac{dy}{dx} = \frac{d}{dx} \: {x}^{ \frac{7}{2} } }$

$\bold{ = \frac{7}{2} {x}^{ \frac{7}{2} - 1 } \: \: ( \because \: \: \frac{d}{dx} {x}^{n} = n {x}^{n - 1} })</p><p>$

$\bold{ = \frac{7}{2} {x}^{ \frac{7 - 2}{2} } }</p><p>$

$\bold{= \frac{7}{2} {x}^{ \frac{5}{2} } }$

$\bold{or \: \: \frac{7 {x}^{ \frac{5}{2} } }{2} }$

$\bold{2. \: \sqrt{35} }$

$\large\pink {\boxed{ \bold{ \green{Solution : }}}}$

$\bold{Suppose \: y = \sqrt{35} }$

Differential with respect to x,

$\bold{ \frac{dy}{dx} = \frac{d}{dx} \sqrt{35} }</p><p>$

$\bold{ = 0}$

$\bold3. \: \: { \sqrt{x}}$

$\large\pink {\boxed{ \bold{ \green{Solution : }}}}$

$\bold{Suppose \: y = \sqrt{x} }$

$\bold{ = {x}^{ \frac{1}{2} } }$

Differential with respect to x,

$\bold{ \frac{dy}{dx} = \frac{d}{dx} \: {x}^{ \frac{1}{2} } }$

$\bold{ = \frac{1}{2} \: {x}^{ \frac{1}{2} - 1 } \: \: { (\because \: \: \frac{d}{dx} {x}^{n} = n {x}^{n - 1} })}$

$\bold { = \frac{1}{2} {x}^{ \frac{1 - 2}{2} } }$

$\bold{ = \frac{1}{2} ( {x})^{ - \frac{1}{2} } }$

$\bold{ = \frac{1}{2 {x}^{ \frac{1}{2} } } }$

$\bold{ = \frac{1}{2 \sqrt{x} } }$

$\bold{ 4.\: \: \: {x}^{ - 3} }$

$\large\pink {\boxed{ \bold{ \green{Solution : }}}}$

$\bold{Suppose \: y = {x}^{ - 3} }$[/tex]

Differential with respect to the x,

$\bold{ = - 3 {x}^{- 3} \: \: { (\because \: \: \frac{d}{dx} {x}^{n} = n {x}^{n - 1} })}$

$\bold{or \: \: \frac{ - 3}{ {x}^{3} } }$

$\bold{6. \: \: \sqrt[3]{x} }$

$\large\pink {\boxed{ \bold{ \green{Solution : }}}}$

$\bold{Suppose \: y = \sqrt[3]{x} }$

$\bold{y = {x}^{ \frac{1}{3} } }$

Differential with respect to the x,

$\bold{ \frac{dy}{dx} = \frac{d}{dx} {x}^{ \frac{1}{3} } }$

$\bold{ = \frac{1}{3} {x}^{ \frac{1}{3} + 1} \: \: { (\because \: \: \frac{d}{dx} {x}^{n} = n {x}^{n - 1} })}$

$\bold{ = \frac{1}{3} {x}^{ \frac{1 + 3}{3} } }$

$\bold{ = \frac{1}{3} {x}^{ \frac{4}{3} } }$

$\bold{7. \: \: 3 {x}^{2} }$

$\large\pink {\boxed{ \bold{ \green{Solution : }}}}$

$\bold{Suppose \: y = 3 {x}^{2} }$

Differential with respect to the x,

$\bold{ \frac{dy}{dx} = \frac{d}{dx} 3 {x}^{2} }$

$\bold{ = 3 \frac{d}{dx} {x}^{2} }$

$\bold{ = 3 \times 2 {x}^{2 + 1} \: \: { (\because \: \: \frac{d}{dx} {x}^{n} = n {x}^{n - 1} })}$

$\bold{ = 6 {x}^{3} }$

$\bold{ 8. \: \: {x}^{10} }$

$\large\pink {\boxed{ \bold{ \green{Solution : }}}}$

$\bold{Suppose \: y = {x}^{10} }$

Differential with respect to the x,

$\bold{ \frac{dy}{dx} = \frac{d}{dx} {x}^{10} }$

$\bold{ = 10 {x}^{10 + 1} \: \: { (\because \: \: \frac{d}{dx} {x}^{n} = n {x}^{n - 1} })}$

$\bold{ = 10 {x}^{11} }$

$\bold{9. \: \: \pi}$

$\large\pink {\boxed{ \bold{ \green{Solution : }}}}$

$\bold{Suppose \: y = \pi }$

$\bold{y = \frac{22}{7} }$

Differential with respect to the x,

$\bold{ \frac{dy}{dx} = \frac{d}{dx } } \frac{22}{7}$

$\bold{ = 0}$

$\bold{10. \: \: {x}^{ \frac{3}{2} } }$

$\large\pink {\boxed{ \bold{ \green{Solution : }}}}$

$\bold{Suppose \: y = {x}^{ \frac{1}{3} } }$

Differential with respect to the x,

$\bold{ \frac{dy}{dx} = \frac{d}{dx} {x}^{ \frac{1}{3} } }$

$\bold{ = \frac{1}{3} {x}^{ \frac{1}{3} + 1} \: \: { (\because \: \: \frac{d}{dx} {x}^{n} = n {x}^{n - 1} })}$

$\bold{ = \frac{1}{3} {x}^{ \frac{1 + 3}{3} } }$

$\bold{ \frac{ {x}^{ \frac{4}{3} } }{3} }$