Find out the answer to "Use the rules for derivatives to find the..."

Chesley

Answered question

2021-10-12

Use the rules for derivatives to find the derivative of each function defined as follows.

y = x \sqrt{x} + \frac{3}{x^{2} \sqrt{x}}

sweererlirumeX

Skilled2021-10-13Added 91 answers

Step 1
To find the derivative of:

y = x \sqrt{x} + \frac{3}{x^{2} \sqrt{x}}

solution:

y = x \sqrt{x} + \frac{3}{x^{2} \sqrt{x}}

On simplifying further, we get:

y = x \sqrt{x} + \frac{3}{x^{2} \sqrt{x}}

y = x^{\frac{3}{2}} + \frac{3}{x^{\frac{5}{2}}}

(using,

a^{b} a^{c} = a^{b + c}

)

y = x^{\frac{3}{2}} + 3 x^{\frac{- 5}{2}}

(using,

\frac{1}{a^{n}} = a^{- n}

)
differentiating both sides w.r.t x we get:

\frac{d}{d x} (y) = \frac{d}{d x} (x^{\frac{3}{2}} + 3 x^{\frac{- 5}{2}})

\Rightarrow \frac{d y}{d x} = (\frac{3}{2} x^{\frac{3}{2} - 1}) + 3 (- \frac{5}{2} x^{\frac{- 5}{2} - 1})

(using,

\frac{d}{d x} (x^{n}) = n x^{n - 1}

)

\Rightarrow \frac{d y}{d x} = \frac{3}{2} \sqrt{x} - \frac{15}{2} x^{- \frac{7}{2}}

\Rightarrow \frac{d y}{d x} = \frac{3}{2} \sqrt{x} - \frac{15}{2 x^{3} \sqrt{x}}

Step 2
Result:

\frac{d y}{d x} = \frac{3}{2} \sqrt{x} - \frac{15}{2 x^{3} \sqrt{x}}

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