Question

Find the derivative of y = x^3/2.

Find the derivative of y = 1/x^3.
Find the derivative of y = 1/√x.
​

Answer 1

`\quad \huge \quad \quad \boxed{ \tt \:Answer }`

`\qquad \tt \rightarrow \:y = \cfrac{d}{dx} ( {x}^{ (3)/(2) } )= (3)/(2) √(x)`

`\qquad \tt \rightarrow \:y = \cfrac{d}{dx} \bigg( \cfrac{1}{ {x}^(3) } \bigg)= \cfrac{- 3}{ {x}^(4) }`

`\qquad \tt \rightarrow \:y = \cfrac{d}{dx} \bigg( \cfrac{1}{ √(x)^{} } \bigg)= \cfrac{ - 1}{ 2\sqrt{{x}^{ { 3}{} } }}`

____________________________________

`\large \tt Solution \: :`

properties to be used here :

`\qquad \tt \rightarrow \:\cfrac{d}{dx}( {x}^( n ) ) = n \sdot{x}^(n - 1)`

`\large \textsf{Question : 1}`

`\qquad \tt \rightarrow \:y = \cfrac{d}{dx} ( {x}^{ (3)/(2) } )`

`\qquad \tt \rightarrow \:y = (3)/(2) x { }^{ (3)/(2) - 1 }`

`\qquad \tt \rightarrow \:y = (3)/(2) x { }^{ (3 - 2)/(2) }`

`\qquad \tt \rightarrow \:y = (3)/(2) x { }^{ (1)/(2) }`

`\qquad \tt \rightarrow \:y = (3)/(2) √(x)`

`\large \textsf{Question : 2}`

`\qquad \tt \rightarrow \:y = \cfrac{d}{dx} \bigg( \cfrac{1}{ {x}^(3) } \bigg)`

`\qquad \tt \rightarrow \:y = \cfrac{d}{dx} ({ {x}^( - 3) } )`

`\qquad \tt \rightarrow \:y = - 3 { {x}^( - 3 - 1) }`

`\qquad \tt \rightarrow \:y = - 3 { {x}^( - 4) }`

`\qquad \tt \rightarrow \:y = \cfrac{- 3}{ {x}^(4) }`

`\large \textsf{Question : 3}`

`\qquad \tt \rightarrow \:y = \cfrac{d}{dx} \bigg( \cfrac{1}{ √(x)^{} } \bigg)`

`\qquad \tt \rightarrow \:y = \cfrac{d}{dx} \bigg( \cfrac{1}{{x}^{ (1)/(2) } } \bigg)`

`\qquad \tt \rightarrow \:y = \cfrac{d}{dx} ({ {x}^{ - (1)/(2) } } )`

`\qquad \tt \rightarrow \:y = -\cfrac{1}{2} { {x}^{ - (1)/(2) - 1} }`

`\qquad \tt \rightarrow \:y = -\cfrac{1}{2} { {x}^{ ( - 1 - 2)/(2) } }`

`\qquad \tt \rightarrow \:y = -\cfrac{1}{2} { {x}^{ ( - 3)/(2) } }`

`\qquad \tt \rightarrow \:y = \cfrac{ - 1}{ 2{x}^{ ( 3)/(2) } }`

`\qquad \tt \rightarrow \:y = \cfrac{ - 1}{ 2\sqrt{{x}^{ { 3}{} } }}`

Answered by : ❝ AǫᴜᴀWɪᴢ ❞