Calculus Examples

$p^{2} + 2 p + q = 16$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d p} (p^{2} + 2 p + q) = \frac{d}{d p} (16)$

Step 2

Differentiate the left side of the equation.

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Step 2.1

Differentiate.

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Step 2.1.1

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

$\frac{d}{d p} [p^{2}] + \frac{d}{d p} [2 p] + \frac{d}{d p} [q]$

Step 2.1.2

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

$2 p + \frac{d}{d p} [2 p] + \frac{d}{d p} [q]$

Step 2.2

Evaluate $\frac{d}{d p} [2 p]$ .

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Step 2.2.1

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

$2 p + 2 \frac{d}{d p} [p] + \frac{d}{d p} [q]$

Step 2.2.2

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

$2 p + 2 \cdot 1 + \frac{d}{d p} [q]$

Step 2.2.3

Multiply $2$ by $1$ .

$2 p + 2 + \frac{d}{d p} [q]$

Step 2.3

Rewrite $\frac{d}{d p} [q]$ as $q'$ .

$2 p + 2 + q'$

Step 3

Since $16$ is constant with respect to $p$ , the derivative of $16$ with respect to $p$ is $0$ .

$0$

Step 4

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

$2 p + 2 + q' = 0$

Step 5

Move all terms not containing $q'$ to the right side of the equation.

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Step 5.1

Subtract $2 p$ from both sides of the equation.

$2 + q' = - 2 p$

Step 5.2

Subtract $2$ from both sides of the equation.

$q' = - 2 p - 2$

Step 6

Replace $q'$ with $\frac{d q}{d p}$ .

$\frac{d q}{d p} = - 2 p - 2$