Find the derivative.
-$$y = \int _ { x ^ { 10 } } ^ { 0 } \cos \sqrt { t d t }$$
A) $- 10 x ^ { 9 } \cos \left( x ^ { 5 } \right)$
B) $- \sin \left( x ^ { 5 } \right)$
C) $1 - \cos \left( x ^ { 5 } \right)$
D) $10 x ^ { 9 } \cos \left( x ^ { 5 } \right)$

Question

Quizplus · Accepted Answer

The derivative of the integral from $x^{10}$ to 0 of $\cos(\sqrt{t}) dt$ with respect to $x$ can be found using the Fundamental Theorem of Calculus Part 1, which states that if $F(x) = \int_{a(x)}^{b(x)} f(t) dt$, then $F'(x) = f(b(x)) \cdot b'(x) - f(a(x)) \cdot a'(x)$. Here, $a(x) = x^{10}$, $b(x) = 0$, and $f(t) = \cos(\sqrt{t})$. Thus, $F'(x) = 0 - \cos(\sqrt{x^{10}}) \cdot (10x^9) = -10x^9 \cos(x^5)$, since $\sqrt{x^{10}} = x^5$ and the derivative of $x^{10}$ is $10x^9$.

Find the Derivative y=∫x100cos⁡tdty = \int _ { x ^ { 10 } } ^ { 0 } \cos \sqrt { t d t }y=∫x100​costdt​

Find the Derivative $y = \int _ { x ^ { 10 } } ^ { 0 } \cos \sqrt { t d t }$