Explained: "Find the first and second derivatives. y=x33+x22+e−x"

jernplate8

Answered question

2021-06-08

Find the first and second derivatives.

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

timbalemX

Skilled2021-06-09Added 108 answers

Step 1
Given:

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

Step 2
Calculation:
To find the first derivative, we have to differentiate the function with respect to x.

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

,(Using power rule:

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

)

\frac{d y}{d x} = x^{2} + x - e^{- x}

...(1)
Step 3
To find the second derivative, we have to differentiate the equation (1) with respect to x.

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

,(Using power rule:

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

)

\frac{d^{2} y}{{d x}^{2}} = 2 x + 1 + e^{- x}

...(2)

Jeffrey Jordon

Expert2021-11-24Added 2605 answers

Answer is given below (on video)

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