Question

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

Derivatives
Evaluate the following derivatives.
$$\displaystyle{\frac{{{d}}}{{{\left.{d}{x}\right.}}}}{\int_{{{x}}}^{{{5}}}}{{\sin{{w}}}^{{{6}}}{d}}{w}$$

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: $$\displaystyle-{\sin{{\left({x}^{{{6}}}\right)}}}$$