Question

Evaluate the following derivatives. \frac{d}{dx}\int_{x}^{5}\sin w^{6}dw

Derivatives
ANSWERED
asked 2021-02-14
Evaluate the following derivatives.
\(\displaystyle{\frac{{{d}}}{{{\left.{d}{x}\right.}}}}{\int_{{{x}}}^{{{5}}}}{{\sin{{w}}}^{{{6}}}{d}}{w}\)

Answers (1)

2021-02-16
Step 1
We first switch the limits by changing the sign
\(\displaystyle{\frac{{{d}}}{{{\left.{d}{x}\right.}}}}{\int_{{{x}}}^{{{5}}}}{{\sin{{w}}}^{{{6}}}{d}}{w}\)
\(\displaystyle={\frac{{{d}}}{{{\left.{d}{x}\right.}}}}{\left[-{\int_{{{5}}}^{{{x}}}}{{\sin{{w}}}^{{{6}}}{d}}{w}\right]}\)
\(\displaystyle=-{\frac{{{d}}}{{{\left.{d}{x}\right.}}}}{\left[{\int_{{{5}}}^{{{x}}}}{{\sin{{w}}}^{{{6}}}{d}}{w}\right]}\)
Step 2
Then we use the second fundamental theorem of calculus to find the derivative
\(\displaystyle=-{\frac{{{d}}}{{{\left.{d}{x}\right.}}}}{\left[{\int_{{{5}}}^{{{x}}}}{{\sin{{w}}}^{{{6}}}{d}}{w}\right]}\)
\(\displaystyle=-{\sin{{\left({x}^{{{6}}}\right)}}}\)
Answer: \(\displaystyle-{\sin{{\left({x}^{{{6}}}\right)}}}\)
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