Algebra Examples

$x^{2} - 7 x y + y^{3} = C$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d y} (x^{2} - 7 x y + y^{3}) = \frac{d}{d y} (C)$

Step 2

Differentiate the left side of the equation.

Tap for more steps...

Step 2.1

Differentiate.

Tap for more steps...

Step 2.1.1

By the Sum Rule, the derivative of $x^{2} - 7 x y + y^{3}$ with respect to $y$ is $\frac{d}{d y} [x^{2}] + \frac{d}{d y} [- 7 x y] + \frac{d}{d y} [y^{3}]$ .

$\frac{d}{d y} [x^{2}] + \frac{d}{d y} [- 7 x y] + \frac{d}{d y} [y^{3}]$

Step 2.1.2

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

$0 + \frac{d}{d y} [- 7 x y] + \frac{d}{d y} [y^{3}]$

Step 2.2

Evaluate $\frac{d}{d y} [- 7 x y]$ .

Tap for more steps...

Step 2.2.1

Since $- 7 x$ is constant with respect to $y$ , the derivative of $- 7 x y$ with respect to $y$ is $- 7 x \frac{d}{d y} [y]$ .

$0 - 7 x \frac{d}{d y} [y] + \frac{d}{d y} [y^{3}]$

Step 2.2.2

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

$0 - 7 x \cdot 1 + \frac{d}{d y} [y^{3}]$

Step 2.2.3

Multiply $- 7$ by $1$ .

$0 - 7 x + \frac{d}{d y} [y^{3}]$

Step 2.3

Differentiate using the Power Rule which states that $\frac{d}{d y} [y^{n}]$ is $n y^{n - 1}$ where $n = 3$ .

$0 - 7 x + 3 y^{2}$

Step 2.4

Simplify.

Tap for more steps...

Step 2.4.1

Subtract $7 x$ from $0$ .

$- 7 x + 3 y^{2}$

Step 2.4.2

Reorder terms.

$3 y^{2} - 7 x$

Step 3

Rewrite $\frac{d}{d y} [C]$ as $C'$ .

$C'$

Step 4

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

$3 y^{2} - 7 x = C'$

Step 5

Rewrite the equation as $C' = 3 y^{2} - 7 x$ .

$C' = 3 y^{2} - 7 x$

Step 6

Replace $C'$ with $\frac{d C}{d y}$ .

$\frac{d C}{d y} = 3 y^{2} - 7 x$