Calculus Examples

$3 \sin (y) + x = y^{2}$ ; $(π^{2}, π)$

Step 1

Differentiate both sides of the equation.

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

Step 2

Differentiate the left side of the equation.

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Step 2.1

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

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

Step 2.2

Evaluate $\frac{d}{d y} [3 \sin (y)]$ .

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Step 2.2.1

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

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

Step 2.2.2

The derivative of $\sin (y)$ with respect to $y$ is $\cos (y)$ .

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

Step 2.3

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

$3 \cos (y) + x'$

Step 2.4

Reorder terms.

$x' + 3 \cos (y)$

Step 3

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

$2 y$

Step 4

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

$x' + 3 \cos (y) = 2 y$

Step 5

Subtract $3 \cos (y)$ from both sides of the equation.

$x' = 2 y - 3 \cos (y)$

Step 6

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

$\frac{d x}{d y} = 2 y - 3 \cos (y)$

Step 7

Replace $x$ with $p^{2}$ and $y$ with $p$ in the expression.

$\frac{d x}{d y} = 2 (p) - 3 \cos (p)$

Step 8

Multiply $2$ by $p$ .

$2 p - 3 \cos (p)$