Question

Evaluate the following derivatives. \frac{d}{dx}\int_{7}^{x}\sqrt{1+t^{4}+t^{6}}dt

Derivatives
ANSWERED
asked 2021-02-22
Evaluate the following derivatives.
\(\displaystyle{\frac{{{d}}}{{{\left.{d}{x}\right.}}}}{\int_{{{7}}}^{{{x}}}}\sqrt{{{1}+{t}^{{{4}}}+{t}^{{{6}}}}}{\left.{d}{t}\right.}\)

Answers (1)

2021-02-24
Step 1
Formula
\(\displaystyle{\frac{{{d}}}{{{\left.{d}{x}\right.}}}}{\int_{{{a}{\left({x}\right)}}}^{{{b}{\left({x}\right)}}}}{f{{\left({x},{t}\right)}}}{\left.{d}{t}\right.}={\int_{{{a}{\left({x}\right)}}}^{{{b}{\left({x}\right)}}}}{\frac{{\partial}}{{\partial{x}}}}{\left({f{{\left({x},{t}\right)}}}{\left.{d}{t}\right.}+{b}'{\left({x}\right)}{f{{\left({x},{b}\right)}}}-{a}'{\left({x}\right)}{f{{\left({x},{a}\right)}}}\right.}\)
Step 2
By formula
\(\displaystyle{\frac{{{d}}}{{{\left.{d}{x}\right.}}}}{\int_{{{7}}}^{{{x}}}}\sqrt{{{1}+{t}^{{{4}}}+{t}^{{{6}}}}}{\left.{d}{t}\right.}={\left({\int_{{{7}}}^{{{x}}}}{0}\right)}+\sqrt{{{1}+{x}^{{{4}}}+{x}^{{{6}}}}}-{0}=\sqrt{{{1}+{x}^{{{4}}}+{x}^{{{6}}}}}\)
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