"Find the derivatives of the functionsy=(1+x2)(x34−x−3)" solved and explained

r1fa8dy5

Answered question

2022-03-25

Find the derivatives of the functions

y = (1 + x^{2}) (x^{\frac{3}{4}} - x^{- 3})

yaum3xg1

Beginner2022-03-26Added 12 answers

Step 1
Given:

y = (1 + x^{2}) (x^{\frac{3}{4}} - x^{- 3})

To Find: The derivative of the given function
Step 2
Solution:

y = (1 + x^{2}) (x^{\frac{3}{4}} - x^{- 3})

Differentiating with respect to x we get

\frac{d y}{d x} = \frac{d}{d x} (1 + x^{2}) (x^{\frac{3}{4}} - x^{- 3})

We know that

\frac{d}{d x} u \cdot v = u \frac{d v}{d x} + v \frac{d u}{d x}

Hence

\frac{d y}{d x} = (1 + x^{2}) \frac{d}{d x} (x^{\frac{3}{4}} - x^{- 3}) + (x^{\frac{3}{4}} - x^{- 3}) \frac{d}{d x} (1 + x^{2})

We also know that

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

Hence

\frac{d y}{d x} = (1 + x^{2}) [\frac{3}{4} x^{(\frac{3}{4} - 1)} - (- 3) x^{(- 3 - 1)}] + (x^{\frac{3}{4}} - x^{- 3}) (0 + 2 x^{(2 - 1)})

\frac{d y}{d x} = (1 + x^{2}) (\frac{3}{4} x^{\frac{- 1}{4}} + 3 x^{- 4}) + (x^{\frac{3}{4}} - x^{- 3}) (2 x)

\frac{d y}{d x} = (1 + x^{2}) (\frac{3}{4} x^{\frac{- 1}{4}} + 3 x^{- 4}) + 2 x (x^{\frac{3}{4}} - x^{- 3})

Hence

\frac{d y}{d x} = (1 + x^{2}) (\frac{3}{4} x^{\frac{- 1}{4}} + 3 x^{- 4}) + 2 x (x^{\frac{3}{4}} - x^{- 3})

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