Calculus Examples

$40 h^{5} + 4000000 = p^{3}$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d p} (40 h^{5} + 4000000) = \frac{d}{d p} (p^{3})$

Step 2

Differentiate the left side of the equation.

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Step 2.1

By the Sum Rule, the derivative of $40 h^{5} + 4000000$ with respect to $p$ is $\frac{d}{d p} [40 h^{5}] + \frac{d}{d p} [4000000]$ .

$\frac{d}{d p} [40 h^{5}] + \frac{d}{d p} [4000000]$

Step 2.2

Evaluate $\frac{d}{d p} [40 h^{5}]$ .

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Step 2.2.1

Since $40$ is constant with respect to $p$ , the derivative of $40 h^{5}$ with respect to $p$ is $40 \frac{d}{d p} [h^{5}]$ .

$40 \frac{d}{d p} [h^{5}] + \frac{d}{d p} [4000000]$

Step 2.2.2

Differentiate using the chain rule, which states that $\frac{d}{d p} [f (g (p))]$ is $f' (g (p)) g' (p)$ where $f (p) = p^{5}$ and $g (p) = h$ .

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Step 2.2.2.1

To apply the Chain Rule, set $u$ as $h$ .

$40 (\frac{d}{d u} [u^{5}] \frac{d}{d p} [h]) + \frac{d}{d p} [4000000]$

Step 2.2.2.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = 5$ .

$40 (5 u^{4} \frac{d}{d p} [h]) + \frac{d}{d p} [4000000]$

Step 2.2.2.3

Replace all occurrences of $u$ with $h$ .

$40 (5 h^{4} \frac{d}{d p} [h]) + \frac{d}{d p} [4000000]$

Step 2.2.3

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

$40 (5 h^{4} h') + \frac{d}{d p} [4000000]$

Step 2.2.4

Multiply $5$ by $40$ .

$200 h^{4} h' + \frac{d}{d p} [4000000]$

Step 2.3

Differentiate using the Constant Rule.

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Step 2.3.1

Since $4000000$ is constant with respect to $p$ , the derivative of $4000000$ with respect to $p$ is $0$ .

$200 h^{4} h' + 0$

Step 2.3.2

Add $200 h^{4} h'$ and $0$ .

$200 h^{4} h'$

Step 3

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

$3 p^{2}$

Step 4

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

$200 h^{4} h' = 3 p^{2}$

Step 5

Divide each term in $200 h^{4} h' = 3 p^{2}$ by $200 h^{4}$ and simplify.

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Step 5.1

Divide each term in $200 h^{4} h' = 3 p^{2}$ by $200 h^{4}$ .

$\frac{200 h^{4} h'}{200 h^{4}} = \frac{3 p^{2}}{200 h^{4}}$

Step 5.2

Simplify the left side.

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Step 5.2.1

Cancel the common factor of $200$ .

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Step 5.2.1.1

Cancel the common factor.

$\frac{200 h^{4} h'}{200 h^{4}} = \frac{3 p^{2}}{200 h^{4}}$

Step 5.2.1.2

Rewrite the expression.

$\frac{h^{4} h'}{h^{4}} = \frac{3 p^{2}}{200 h^{4}}$

Step 5.2.2

Cancel the common factor of $h^{4}$ .

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Step 5.2.2.1

Cancel the common factor.

$\frac{h^{4} h'}{h^{4}} = \frac{3 p^{2}}{200 h^{4}}$

Step 5.2.2.2

Divide $h'$ by $1$ .

$h' = \frac{3 p^{2}}{200 h^{4}}$

Step 6

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

$\frac{d h}{d p} = \frac{3 p^{2}}{200 h^{4}}$