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(Solved): \[ \sin (x y)=\cos (x+y) \] , then \( y^{\prime}=( \) \[ \text { C }-\frac{y \cos (x y)+\sin (x+y)} ...
![\[
\sin (x y)=\cos (x+y)
\]
, then \( y^{\prime}=( \)
\[
\text { C }-\frac{y \cos (x y)+\sin (x+y)}{x \cos (x y)+\sin (x+y)}](https://media.cheggcdn.com/study/db2/db272c00-4f83-4450-a102-59dde4c13ae8/image)
![Single Choice (total7question, 28.0score)
6. (4.0score)
\[
\int x e^{-x^{2}} d x=()
\]
\[
\text { C }-\frac{1}{2} e^{-x^{2}}+](https://media.cheggcdn.com/study/a32/a3278085-3f08-4560-a935-d32344ac721d/image)
![Fill in the Blank (total9question,
36.0score)
8. (4.0score)
\[
\lim _{x \rightarrow 1} \frac{x^{2}-1}{x-1}
\]](https://media.cheggcdn.com/study/b62/b627ae16-c193-420d-a4bc-37774b20b22c/image)
\[ \sin (x y)=\cos (x+y) \] , then \( y^{\prime}=( \) \[ \text { C }-\frac{y \cos (x y)+\sin (x+y)}{x \cos (x y)+\sin (x+y)} \]
Single Choice (total7question, 28.0score) 6. (4.0score) \[ \int x e^{-x^{2}} d x=() \] \[ \text { C }-\frac{1}{2} e^{-x^{2}}+C \text {. } \]
Single Choice (total7question, 28.0score) 7. (4.0score) \( \int \cos 2 x d x=(\quad) \) D \( \frac{1}{2} \sin 2 x+C \).
Fill in the Blank (total9question, 36.0score) 8. (4.0score) \[ \lim _{x \rightarrow 1} \frac{x^{2}-1}{x-1} \]
Expert Answer
Solution- Given that sin?(xy)=cos?(x+y) we find y?=? Differentating with respect to x on both the sides we have cos?(xy)[y+xy?]=?sin?(x+y)[1+y?] ?ycos