Final Answer
$y^{\prime}=\frac{-x}{y}$
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Step-by-step Solution
Problem to solve:
$\frac{d}{dx}\left(x^2+y^2=25\right)$
Specify the solving method
1
Apply implicit differentiation by taking the derivative of both sides of the equation with respect to the differentiation variable
$\frac{d}{dx}\left(x^2+y^2\right)=\frac{d}{dx}\left(25\right)$
2
The derivative of the constant function ($25$) is equal to zero
$\frac{d}{dx}\left(x^2+y^2\right)=0$
3 Try to guess Step 3. Or become premium for the price of a latte.
4
The power rule for differentiation states that if $n$ is a real number and $f(x) = x^n$, then $f'(x) = nx^{n-1}$
$\frac{d}{dx}\left(x^2\right)+2y\frac{d}{dx}\left(y\right)=0$
$2\cdot 1y$
Any expression multiplied by $1$ is equal to itself
$2y$
5
The derivative of the linear function is equal to $1$
$\frac{d}{dx}\left(x^2\right)+2y\cdot y^{\prime}=0$
6 Try to guess Step 6. Or become premium for the price of a latte.
7
Isolate $2y\cdot y^{\prime}$
$2y\cdot y^{\prime}=0-2x$
8
$x+0=x$, where $x$ is any expression
$2y\cdot y^{\prime}=-2x$
9 Try to guess Step 9. Or become premium for the price of a latte.
10
Simplifying the quotients
$y\cdot y^{\prime}=\frac{-2x}{2}$
11
Take $\frac{-2}{2}$ out of the fraction
$y\cdot y^{\prime}=-x$
12 Try to guess Step 12. Or become premium for the price of a latte.
13
Simplifying the quotients
$y^{\prime}=\frac{-x}{y}$
Final Answer
$y^{\prime}=\frac{-x}{y}$