Question

Find the first and second derivatives. y=-x^{2}+3

Derivatives
ANSWERED
asked 2021-05-04
Find the first and second derivatives.
\(\displaystyle{y}=-{x}^{{{2}}}+{3}\)

Answers (1)

2021-05-05
Step 1
The given function is \(\displaystyle{y}=-{x}^{{{2}}}+{3}\).
Formula:
\(\displaystyle{\frac{{{d}}}{{{\left.{d}{x}\right.}}}}{\left({x}^{{{n}}}\right)}={n}{x}^{{{n}-{1}}}\)
\(\displaystyle{\frac{{{d}}}{{{\left.{d}{x}\right.}}}}{\left({f}+{g}\right)}={\frac{{{d}{f}}}{{{\left.{d}{x}\right.}}}}+{\frac{{{d}{g}}}{{{\left.{d}{x}\right.}}}}\)
\(\displaystyle{\frac{{{d}}}{{{\left.{d}{x}\right.}}}}{\left({k}\right)}={0}\) where k is constant.
\(\displaystyle{\frac{{{d}}}{{{\left.{d}{x}\right.}}}}{\left({k}{f}\right)}={k}{\frac{{{d}}}{{{\left.{d}{x}\right.}}}}{\left({f}\right)}\)
Step 2
Obtain the first derivative of y as follows.
\(\displaystyle{y}=-{x}^{{{2}}}+{3}\)
\(\displaystyle{\frac{{{\left.{d}{y}\right.}}}{{{\left.{d}{x}\right.}}}}={\frac{{{d}}}{{{\left.{d}{x}\right.}}}}{\left(-{x}^{{{2}}}+{3}\right)}\)
\(\displaystyle={\frac{{{d}}}{{{\left.{d}{x}\right.}}}}{\left(-{x}^{{{2}}}\right)}+{\frac{{{d}}}{{{\left.{d}{x}\right.}}}}{\left({3}\right)}\)
\(\displaystyle=-{\frac{{{d}}}{{{\left.{d}{x}\right.}}}}{\left({x}^{{{2}}}\right)}+{\frac{{{d}}}{{{\left.{d}{x}\right.}}}}{\left({3}\right)}\)
\(\displaystyle=-{2}{x}^{{{2}-{1}}}+{0}\)
\(\displaystyle=-{2}{x}^{{{1}}}\)
=-2x
Step 3
Obtain the second derivative as follows.
\(\displaystyle{\frac{{{\left.{d}{y}\right.}}}{{{\left.{d}{x}\right.}}}}=-{2}{x}\)
\(\displaystyle{\frac{{{d}^{{{2}}}{y}}}{{{\left.{d}{x}\right.}^{{{2}}}}}}={\frac{{{d}}}{{{\left.{d}{x}\right.}}}}{\left({\frac{{{\left.{d}{y}\right.}}}{{{\left.{d}{x}\right.}}}}\right)}\)
\(\displaystyle={\frac{{{d}}}{{{\left.{d}{x}\right.}}}}{\left(-{2}{x}\right)}\)
\(\displaystyle=-{2}{\frac{{{d}}}{{{\left.{d}{x}\right.}}}}{\left({x}\right)}\)
\(\displaystyle=-{2}\times{1}{x}^{{{1}-{1}}}\)
\(\displaystyle=-{2}{x}^{{{0}}}\)
=-2
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