Find da/dx x^2+y^2=a^2

$x^{2} + y^{2} = a^{2}$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (x^{2} + y^{2}) = \frac{d}{d x} (a^{2})$

Step 2

Differentiate the left side of the equation.

By the Sum Rule, the derivative of $x^{2} + y^{2}$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [y^{2}]$ .

$\frac{d}{d x} [x^{2}] + \frac{d}{d x} [y^{2}]$

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$2 x + \frac{d}{d x} [y^{2}]$

Since $y^{2}$ is constant with respect to $x$ , the derivative of $y^{2}$ with respect to $x$ is $0$ .

$2 x + 0$

Add $2 x$ and $0$ .

$2 x$

Step 3

Differentiate the right side of the equation.

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{2}$ and $g (x) = a$ .

To apply the Chain Rule, set $u$ as $a$ .

$\frac{d}{d u} [u^{2}] \frac{d}{d x} [a]$

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = 2$ .

$2 u \frac{d}{d x} [a]$

Replace all occurrences of $u$ with $a$ .

$2 a \frac{d}{d x} [a]$

Rewrite $\frac{d}{d x} [a]$ as $a'$ .

$2 a a'$

Step 4

Reform the equation by setting the left side equal to the right side.

$2 x = 2 a a'$

Step 5

Solve for $a'$ .

Rewrite the equation as $2 a a' = 2 x$ .

$2 a a' = 2 x$

Divide each term in $2 a a' = 2 x$ by $2 a$ and simplify.

Divide each term in $2 a a' = 2 x$ by $2 a$ .

$\frac{2 a a'}{2 a} = \frac{2 x}{2 a}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 a a'}{2 a} = \frac{2 x}{2 a}$

Rewrite the expression.

$\frac{a a'}{a} = \frac{2 x}{2 a}$

Cancel the common factor of $a$ .

Cancel the common factor.

$\frac{a a'}{a} = \frac{2 x}{2 a}$

Divide $a'$ by $1$ .

$a' = \frac{2 x}{2 a}$

Simplify the right side.

Cancel the common factor of $2$ .

Cancel the common factor.

$a' = \frac{2 x}{2 a}$

Rewrite the expression.

$a' = \frac{x}{a}$

Step 6

Replace $a'$ with $\frac{d a}{d x}$ .

$\frac{d a}{d x} = \frac{x}{a}$

Find da/dx x^2+y^2=a^2

Related Questions

Support LCMGCF.com by disabling your adblocker.