x^{2} +y^{2} =25 What is dydx dx/dy?

Question
Differential equations
$$x^{2} +y^{2} =25$$
What is dydx dx/dy?

2021-02-10
$$\frac{dx}{dy} =−\frac{x}{y}$$
Since y is not on it's own side, we will have to differentiate implicitly. We will have to take the derivative with respect to x, since the question is asking for dydx dx/dy . After taking the derivative we get: $$2x+2y(dydx)=02x+2y\frac{dx}{dy}=0$$
Since we want to solve for dy/dx we must isolate it. First, we will subtract 2x from both sides to get: $$2y(dydx)=−2x2y\frac{dx}{dy}=−2x$$
Then we will divide both sides by 2y to get: $$dy/dx=−\frac{x}{y}$$
$$\frac{dx}{dy} =−\frac{x}{y}$$

Relevant Questions

Solve the equation:
$$\displaystyle{\left({x}+{1}\right)}{\frac{{{\left.{d}{y}\right.}}}{{{\left.{d}{x}\right.}}}}={x}{\left({y}^{{2}}+{1}\right)}$$
Solve the equation:
$$\displaystyle{\left({a}-{x}\right)}{\left.{d}{y}\right.}+{\left({a}+{y}\right)}{\left.{d}{x}\right.}={0}$$
The graph of y = f(x) contains the point (0,2), $$\displaystyle{\frac{{{\left.{d}{y}\right.}}}{{{\left.{d}{x}\right.}}}}={\frac{{-{x}}}{{{y}{e}^{{{x}^{{2}}}}}}}$$, and f(x) is greater than 0 for all x, then f(x)=
A) $$\displaystyle{3}+{e}^{{-{x}^{{2}}}}$$
B) $$\displaystyle\sqrt{{{3}}}+{e}^{{-{x}}}$$
C) $$\displaystyle{1}+{e}^{{-{x}}}$$
D) $$\displaystyle\sqrt{{{3}+{e}^{{-{x}^{{2}}}}}}$$
E) $$\displaystyle\sqrt{{{3}+{e}^{{{x}^{{2}}}}}}$$
Consider the curves in the first quadrant that have equationsy=Aexp(7x), where A is a positive constant. Different valuesof A give different curves. The curves form a family,F. Let P=(6,6). Let C be the number of the family Fthat goes through P.
A. Let y=f(x) be the equation of C. Find f(x).
B. Find the slope at P of the tangent to C.
C. A curve D is a perpendicular to C at P. What is the slope of thetangent to D at the point P?
D. Give a formula g(y) for the slope at (x,y) of the member of Fthat goes through (x,y). The formula should not involve A orx.
E. A curve which at each of its points is perpendicular to themember of the family F that goes through that point is called anorthogonal trajectory of F. Each orthogonal trajectory to Fsatisfies the differential equation dy/dx = -1/g(y), where g(y) isthe answer to part D.
Find a function of h(y) such that x=h(y) is the equation of theorthogonal trajectory to F that passes through the point P.
Q. 2# $$(x+1)\frac{dy}{dx}=x(y^{2}+1)$$
Q. 2# $$\displaystyle{\left({x}+{1}\right)}{\frac{{{\left.{d}{y}\right.}}}{{{\left.{d}{x}\right.}}}}={x}{\left({y}^{{{2}}}+{1}\right)}$$
$$\displaystyle{y}={\frac{{{1}}}{{{2}+{\sin{{x}}}}}}$$
find general solution in semi homogenous method of $$\displaystyle\frac{{\left.{d}{y}\right.}}{{\left.{d}{x}\right.}}={x}-{y}+\frac{{1}}{{x}}+{y}-{1}$$
Deterrmine the first derivative $$\displaystyle{\left(\frac{{\left.{d}{y}\right.}}{{\left.{d}{x}\right.}}\right)}$$ :
$$\displaystyle{y}={2}{e}^{{2}}{x}+{I}{n}{x}^{{3}}-{2}{e}^{{x}}$$