Calculus Examples

$p = \frac{t + 1770}{50 (t + 2)}$

Step 1

Differentiate using the Quotient Rule which states that $\frac{d}{d p} [\frac{f (p)}{g (p)}]$ is $\frac{g (p) \frac{d}{d p} [f (p)] - f (p) \frac{d}{d p} [g (p)]}{{g (p)}^{2}}$ where $f (p) = t + 1770$ and $g (p) = 50 (t + 2)$ .

$\frac{50 (t + 2) \frac{d}{d p} [t + 1770] - (t + 1770) \frac{d}{d p} [50 (t + 2)]}{{(50 (t + 2))}^{2}}$

Step 2

Differentiate.

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Step 2.1

By the Sum Rule, the derivative of $t + 1770$ with respect to $p$ is $\frac{d}{d p} [t] + \frac{d}{d p} [1770]$ .

$\frac{50 (t + 2) (\frac{d}{d p} [t] + \frac{d}{d p} [1770]) - (t + 1770) \frac{d}{d p} [50 (t + 2)]}{{(50 (t + 2))}^{2}}$

Step 2.2

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

$\frac{50 (t + 2) (0 + \frac{d}{d p} [1770]) - (t + 1770) \frac{d}{d p} [50 (t + 2)]}{{(50 (t + 2))}^{2}}$

Step 2.3

Since $1770$ is constant with respect to $p$ , the derivative of $1770$ with respect to $p$ is $0$ .

$\frac{50 (t + 2) (0 + 0) - (t + 1770) \frac{d}{d p} [50 (t + 2)]}{{(50 (t + 2))}^{2}}$

Step 2.4

Add $0$ and $0$ .

$\frac{50 (t + 2) \cdot 0 - (t + 1770) \frac{d}{d p} [50 (t + 2)]}{{(50 (t + 2))}^{2}}$

Step 2.5

Since $50 (t + 2)$ is constant with respect to $p$ , the derivative of $50 (t + 2)$ with respect to $p$ is $0$ .

$\frac{50 (t + 2) \cdot 0 - (t + 1770) \cdot 0}{{(50 (t + 2))}^{2}}$

Step 3

Simplify.

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Step 3.1

Apply the product rule to $50 (t + 2)$ .

$\frac{50 (t + 2) \cdot 0 - (t + 1770) \cdot 0}{50^{2} {(t + 2)}^{2}}$

Step 3.2

Apply the distributive property.

$\frac{(50 t + 50 \cdot 2) \cdot 0 - (t + 1770) \cdot 0}{50^{2} {(t + 2)}^{2}}$

Step 3.3

Apply the distributive property.

$\frac{50 t \cdot 0 + 50 \cdot 2 \cdot 0 - (t + 1770) \cdot 0}{50^{2} {(t + 2)}^{2}}$

Step 3.4

Apply the distributive property.

$\frac{50 t \cdot 0 + 50 \cdot 2 \cdot 0 + (- t - 1 \cdot 1770) \cdot 0}{50^{2} {(t + 2)}^{2}}$

Step 3.5

Apply the distributive property.

$\frac{50 t \cdot 0 + 50 \cdot 2 \cdot 0 - t \cdot 0 - 1 \cdot 1770 \cdot 0}{50^{2} {(t + 2)}^{2}}$

Step 3.6

Simplify the numerator.

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Step 3.6.1

Simplify each term.

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Step 3.6.1.1

Multiply $50 t \cdot 0$ .

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Step 3.6.1.1.1

Multiply $0$ by $50$ .

$\frac{0 t + 50 \cdot 2 \cdot 0 - t \cdot 0 - 1 \cdot 1770 \cdot 0}{50^{2} {(t + 2)}^{2}}$

Step 3.6.1.1.2

Multiply $0$ by $t$ .

$\frac{0 + 50 \cdot 2 \cdot 0 - t \cdot 0 - 1 \cdot 1770 \cdot 0}{50^{2} {(t + 2)}^{2}}$

Step 3.6.1.2

Multiply $50 \cdot 2 \cdot 0$ .

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Step 3.6.1.2.1

Multiply $50$ by $2$ .

$\frac{0 + 100 \cdot 0 - t \cdot 0 - 1 \cdot 1770 \cdot 0}{50^{2} {(t + 2)}^{2}}$

Step 3.6.1.2.2

Multiply $100$ by $0$ .

$\frac{0 + 0 - t \cdot 0 - 1 \cdot 1770 \cdot 0}{50^{2} {(t + 2)}^{2}}$

Step 3.6.1.3

Multiply $- t \cdot 0$ .

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Step 3.6.1.3.1

Multiply $0$ by $- 1$ .

$\frac{0 + 0 + 0 t - 1 \cdot 1770 \cdot 0}{50^{2} {(t + 2)}^{2}}$

Step 3.6.1.3.2

Multiply $0$ by $t$ .

$\frac{0 + 0 + 0 - 1 \cdot 1770 \cdot 0}{50^{2} {(t + 2)}^{2}}$

Step 3.6.1.4

Multiply $- 1 \cdot 1770 \cdot 0$ .

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Step 3.6.1.4.1

Multiply $- 1$ by $1770$ .

$\frac{0 + 0 + 0 - 1770 \cdot 0}{50^{2} {(t + 2)}^{2}}$

Step 3.6.1.4.2

Multiply $- 1770$ by $0$ .

$\frac{0 + 0 + 0 + 0}{50^{2} {(t + 2)}^{2}}$

Step 3.6.2

Add $0$ and $0$ .

$\frac{0 + 0 + 0}{50^{2} {(t + 2)}^{2}}$

Step 3.6.3

Add $0$ and $0$ .

$\frac{0 + 0}{50^{2} {(t + 2)}^{2}}$

Step 3.6.4

Add $0$ and $0$ .

$\frac{0}{50^{2} {(t + 2)}^{2}}$

Step 3.7

Combine terms.

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Step 3.7.1

Raise $50$ to the power of $2$ .

$\frac{0}{2500 {(t + 2)}^{2}}$

Step 3.7.2

Cancel the common factor of $0$ and $2500$ .

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Step 3.7.2.1

Factor $2500$ out of $0$ .

$\frac{2500 (0)}{2500 {(t + 2)}^{2}}$

Step 3.7.2.2

Cancel the common factors.

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Step 3.7.2.2.1

Factor $2500$ out of $2500 {(t + 2)}^{2}$ .

$\frac{2500 (0)}{2500 ({(t + 2)}^{2})}$

Step 3.7.2.2.2

Cancel the common factor.

$\frac{2500 \cdot 0}{2500 {(t + 2)}^{2}}$

Step 3.7.2.2.3

Rewrite the expression.

$\frac{0}{{(t + 2)}^{2}}$

Step 3.8

Divide $0$ by ${(t + 2)}^{2}$ .

$0$