Question

instructions

• work this problem out on 1 sheet of paper, one - side only. label the problem, \implicit differentiation\
• you will make a pdf of your all of your work - outs after you complete the test;
• make this problem page 2.
• after you submit this test, then submit your pdf in the canvas portal titled, \exam 2 work - out questions\
• an instructor will grade your work - out questions.

work - out problem 2, 10 points
find $\frac{dy}{dx}$ by implicit differentiation: $ycos x=x^{2}+y^{3}$.
click to proceed when you have completed the work on your own paper.

Explanation:

Step1: Differentiate both sides

Differentiate $y\cos x$ and $x^{2}+y^{2}$ with respect to $x$. Using the product - rule $(uv)^\prime = u^\prime v+uv^\prime$ for $y\cos x$ where $u = y$ and $v=\cos x$. The derivative of $y\cos x$ is $y^\prime\cos x - y\sin x$, and the derivative of $x^{2}+y^{2}$ is $2x + 2yy^\prime$. So we have $y^\prime\cos x - y\sin x=2x + 2yy^\prime$.

Step2: Isolate $y^\prime$

Rearrange the terms to get all terms with $y^\prime$ on one side: $y^\prime\cos x-2yy^\prime=2x + y\sin x$. Factor out $y^\prime$: $y^\prime(\cos x - 2y)=2x + y\sin x$.

Step3: Solve for $y^\prime$

Divide both sides by $\cos x - 2y$ to find $y^\prime$. So $y^\prime=\frac{2x + y\sin x}{\cos x - 2y}$.

Answer:

$\frac{2x + y\sin x}{\cos x - 2y}$