Calculus Examples

$3 y^{2} + x = 27$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d y} (3 y^{2} + x) = \frac{d}{d y} (27)$

Step 2

Differentiate the left side of the equation.

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Step 2.1

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

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

Step 2.2

Evaluate $\frac{d}{d y} [3 y^{2}]$ .

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Step 2.2.1

Since $3$ is constant with respect to $y$ , the derivative of $3 y^{2}$ with respect to $y$ is $3 \frac{d}{d y} [y^{2}]$ .

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

Step 2.2.2

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

$3 (2 y) + \frac{d}{d y} [x]$

Step 2.2.3

Multiply $2$ by $3$ .

$6 y + \frac{d}{d y} [x]$

Step 2.3

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

$6 y + x'$

Step 3

Since $27$ is constant with respect to $y$ , the derivative of $27$ with respect to $y$ is $0$ .

$0$

Step 4

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

$6 y + x' = 0$

Step 5

Subtract $6 y$ from both sides of the equation.

$x' = - 6 y$

Step 6

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

$\frac{d x}{d y} = - 6 y$