Question

Find the first and second derivatives. y=\frac{7x^{5}}{5}-2x+9e^{x} \frac{dy}{dx}=

Derivatives
ANSWERED
asked 2021-06-08
Find the first and second derivatives.
\(\displaystyle{y}={\frac{{{7}{x}^{{{5}}}}}{{{5}}}}-{2}{x}+{9}{e}^{{{x}}}\)
\(\displaystyle{\frac{{{\left.{d}{y}\right.}}}{{{\left.{d}{x}\right.}}}}=\)

Answers (1)

2021-06-09
Step 1
The given function is \(\displaystyle{y}={\frac{{{7}{x}^{{{5}}}}}{{{5}}}}-{2}{x}+{9}{e}^{{{x}}}\)
Obtain the first derivative as follows.
\(\displaystyle{\frac{{{\left.{d}{y}\right.}}}{{{\left.{d}{x}\right.}}}}={\frac{{{d}}}{{{\left.{d}{x}\right.}}}}{\left({\frac{{{7}{x}^{{{5}}}}}{{{5}}}}-{2}{x}+{9}{e}^{{{x}}}\right)}\)
\(\displaystyle={\frac{{{35}{x}^{{{4}}}}}{{{5}}}}-{2}+{9}{e}^{{{x}}}\)
\(\displaystyle={7}{x}^{{{4}}}-{2}+{9}{e}^{{{x}}}\)
Step 2
Obtain the second derivative as follows.
\(\displaystyle{\frac{{{d}^{{{2}}}{y}}}{{{\left.{d}{x}\right.}^{{{2}}}}}}={\frac{{{d}}}{{{\left.{d}{x}\right.}}}}{\left({\frac{{{\left.{d}{y}\right.}}}{{{\left.{d}{x}\right.}}}}\right)}\)
\(\displaystyle{\frac{{{d}}}{{{\left.{d}{x}\right.}}}}{\left({7}{x}^{{{4}}}-{2}+{9}{e}^{{{x}}}\right)}\)
\(\displaystyle={28}{x}^{{{3}}}+{9}{e}^{{{x}}}\)
Thus, the first derivative is \(\displaystyle{\frac{{{\left.{d}{y}\right.}}}{{{\left.{d}{x}\right.}}}}={7}{x}^{{{4}}}-{2}+{9}{e}^{{{x}}}\)
The second derivative is \(\displaystyle{\frac{{{d}^{{{2}}}{y}}}{{{\left.{d}{x}\right.}^{{{2}}}}}}={28}{x}^{{{3}}}+{9}{e}^{{{x}}}\)
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