Algebra Examples

$z = (u^{3} - v^{3}) e^{(u^{3} + v^{3})}$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d z} (z) = \frac{d}{d z} ((u^{3} - {u'}^{3}) e^{u^{3} + {u'}^{3}})$

Step 2

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

$1$

Step 3

Differentiate the right side of the equation.

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Step 3.1

Differentiate using the Product Rule which states that $\frac{d}{d z} [f (z) g (z)]$ is $f (z) \frac{d}{d z} [g (z)] + g (z) \frac{d}{d z} [f (z)]$ where $f (z) = u^{3} - {u'}^{3}$ and $g (z) = e^{u^{3} + {u'}^{3}}$ .

$(u^{3} - {u'}^{3}) \frac{d}{d z} [e^{u^{3} + {u'}^{3}}] + e^{u^{3} + {u'}^{3}} \frac{d}{d z} [u^{3} - {u'}^{3}]$

Step 3.2

Differentiate using the chain rule, which states that $\frac{d}{d z} [f (g (z))]$ is $f' (g (z)) g' (z)$ where $f (z) = e^{z}$ and $g (z) = u^{3} + {u'}^{3}$ .

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Step 3.2.1

To apply the Chain Rule, set $u_{1}$ as $u^{3} + {u'}^{3}$ .

$(u^{3} - {u'}^{3}) (\frac{d}{d u_{1}} [e^{u_{1}}] \frac{d}{d z} [u^{3} + {u'}^{3}]) + e^{u^{3} + {u'}^{3}} \frac{d}{d z} [u^{3} - {u'}^{3}]$

Step 3.2.2

Differentiate using the Exponential Rule which states that $\frac{d}{d u_{1}} [a^{u_{1}}]$ is $a^{u_{1}} \ln (a)$ where $a$ = $e$ .

$(u^{3} - {u'}^{3}) (e^{u_{1}} \frac{d}{d z} [u^{3} + {u'}^{3}]) + e^{u^{3} + {u'}^{3}} \frac{d}{d z} [u^{3} - {u'}^{3}]$

Step 3.2.3

Replace all occurrences of $u_{1}$ with $u^{3} + {u'}^{3}$ .

$(u^{3} - {u'}^{3}) (e^{u^{3} + {u'}^{3}} \frac{d}{d z} [u^{3} + {u'}^{3}]) + e^{u^{3} + {u'}^{3}} \frac{d}{d z} [u^{3} - {u'}^{3}]$

Step 3.3

By the Sum Rule, the derivative of $u^{3} + {u'}^{3}$ with respect to $z$ is $\frac{d}{d z} [u^{3}] + \frac{d}{d z} [{u'}^{3}]$ .

$(u^{3} - {u'}^{3}) e^{u^{3} + {u'}^{3}} (\frac{d}{d z} [u^{3}] + \frac{d}{d z} [{u'}^{3}]) + e^{u^{3} + {u'}^{3}} \frac{d}{d z} [u^{3} - {u'}^{3}]$

Step 3.4

Differentiate using the chain rule, which states that $\frac{d}{d z} [f (g (z))]$ is $f' (g (z)) g' (z)$ where $f (z) = z^{3}$ and $g (z) = u$ .

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Step 3.4.1

To apply the Chain Rule, set $u_{2}$ as $u$ .

$(u^{3} - {u'}^{3}) e^{u^{3} + {u'}^{3}} (\frac{d}{d u_{2}} [{u_{2}}^{3}] \frac{d}{d z} [u] + \frac{d}{d z} [{u'}^{3}]) + e^{u^{3} + {u'}^{3}} \frac{d}{d z} [u^{3} - {u'}^{3}]$

Step 3.4.2

Differentiate using the Power Rule which states that $\frac{d}{d u_{2}} [{u_{2}}^{n}]$ is $n {u_{2}}^{n - 1}$ where $n = 3$ .

$(u^{3} - {u'}^{3}) e^{u^{3} + {u'}^{3}} (3 {u_{2}}^{2} \frac{d}{d z} [u] + \frac{d}{d z} [{u'}^{3}]) + e^{u^{3} + {u'}^{3}} \frac{d}{d z} [u^{3} - {u'}^{3}]$

Step 3.4.3

Replace all occurrences of $u_{2}$ with $u$ .

$(u^{3} - {u'}^{3}) e^{u^{3} + {u'}^{3}} (3 u^{2} \frac{d}{d z} [u] + \frac{d}{d z} [{u'}^{3}]) + e^{u^{3} + {u'}^{3}} \frac{d}{d z} [u^{3} - {u'}^{3}]$

Step 3.5

Rewrite $\frac{d}{d z} [u]$ as $u'$ .

$(u^{3} - {u'}^{3}) e^{u^{3} + {u'}^{3}} (3 u^{2} u' + \frac{d}{d z} [{u'}^{3}]) + e^{u^{3} + {u'}^{3}} \frac{d}{d z} [u^{3} - {u'}^{3}]$

Step 3.6

Differentiate.

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Step 3.6.1

Since ${u'}^{3}$ is constant with respect to $z$ , the derivative of ${u'}^{3}$ with respect to $z$ is $0$ .

$(u^{3} - {u'}^{3}) e^{u^{3} + {u'}^{3}} (3 u^{2} u' + 0) + e^{u^{3} + {u'}^{3}} \frac{d}{d z} [u^{3} - {u'}^{3}]$

Step 3.6.2

Add $3 u^{2} u'$ and $0$ .

$(u^{3} - {u'}^{3}) e^{u^{3} + {u'}^{3}} (3 u^{2} u') + e^{u^{3} + {u'}^{3}} \frac{d}{d z} [u^{3} - {u'}^{3}]$

Step 3.6.3

By the Sum Rule, the derivative of $u^{3} - {u'}^{3}$ with respect to $z$ is $\frac{d}{d z} [u^{3}] + \frac{d}{d z} [- {u'}^{3}]$ .

$(u^{3} - {u'}^{3}) e^{u^{3} + {u'}^{3}} (3 u^{2} u') + e^{u^{3} + {u'}^{3}} (\frac{d}{d z} [u^{3}] + \frac{d}{d z} [- {u'}^{3}])$

Step 3.7

Differentiate using the chain rule, which states that $\frac{d}{d z} [f (g (z))]$ is $f' (g (z)) g' (z)$ where $f (z) = z^{3}$ and $g (z) = u$ .

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Step 3.7.1

To apply the Chain Rule, set $u_{3}$ as $u$ .

$(u^{3} - {u'}^{3}) e^{u^{3} + {u'}^{3}} (3 u^{2} u') + e^{u^{3} + {u'}^{3}} (\frac{d}{d u_{3}} [{u_{3}}^{3}] \frac{d}{d z} [u] + \frac{d}{d z} [- {u'}^{3}])$

Step 3.7.2

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

$(u^{3} - {u'}^{3}) e^{u^{3} + {u'}^{3}} (3 u^{2} u') + e^{u^{3} + {u'}^{3}} (3 {u_{3}}^{2} \frac{d}{d z} [u] + \frac{d}{d z} [- {u'}^{3}])$

Step 3.7.3

Replace all occurrences of $u_{3}$ with $u$ .

$(u^{3} - {u'}^{3}) e^{u^{3} + {u'}^{3}} (3 u^{2} u') + e^{u^{3} + {u'}^{3}} (3 u^{2} \frac{d}{d z} [u] + \frac{d}{d z} [- {u'}^{3}])$

Step 3.8

Rewrite $\frac{d}{d z} [u]$ as $u'$ .

$(u^{3} - {u'}^{3}) e^{u^{3} + {u'}^{3}} (3 u^{2} u') + e^{u^{3} + {u'}^{3}} (3 u^{2} u' + \frac{d}{d z} [- {u'}^{3}])$

Step 3.9

Since $- {u'}^{3}$ is constant with respect to $z$ , the derivative of $- {u'}^{3}$ with respect to $z$ is $0$ .

$(u^{3} - {u'}^{3}) e^{u^{3} + {u'}^{3}} (3 u^{2} u') + e^{u^{3} + {u'}^{3}} (3 u^{2} u' + 0)$

Step 3.10

Add $3 u^{2} u'$ and $0$ .

$(u^{3} - {u'}^{3}) e^{u^{3} + {u'}^{3}} (3 u^{2} u') + e^{u^{3} + {u'}^{3}} (3 u^{2} u')$

Step 3.11

Simplify.

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Step 3.11.1

Apply the distributive property.

$(u^{3} e^{u^{3} + {u'}^{3}} - {u'}^{3} e^{u^{3} + {u'}^{3}}) (3 u^{2} u') + e^{u^{3} + {u'}^{3}} (3 u^{2} u')$

Step 3.11.2

Apply the distributive property.

$u^{3} e^{u^{3} + {u'}^{3}} (3 u^{2} u') - {u'}^{3} e^{u^{3} + {u'}^{3}} (3 u^{2} u') + e^{u^{3} + {u'}^{3}} (3 u^{2} u')$

Step 3.11.3

Combine terms.

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Step 3.11.3.1

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$u^{2 + 3} e^{u^{3} + {u'}^{3}} (3 u') - {u'}^{3} e^{u^{3} + {u'}^{3}} (3 u^{2} u') + e^{u^{3} + {u'}^{3}} (3 u^{2} u')$

Step 3.11.3.2

Add $2$ and $3$ .

$u^{5} e^{u^{3} + {u'}^{3}} (3 u') - {u'}^{3} e^{u^{3} + {u'}^{3}} (3 u^{2} u') + e^{u^{3} + {u'}^{3}} (3 u^{2} u')$

Step 3.11.3.3

Multiply $3$ by $- 1$ .

$u^{5} e^{u^{3} + {u'}^{3}} (3 u') - 3 {u'}^{3} e^{u^{3} + {u'}^{3}} (u^{2} u') + e^{u^{3} + {u'}^{3}} (3 u^{2} u')$

Step 3.11.4

Reorder terms.

$3 e^{u^{3} + {u'}^{3}} u^{5} u' - 3 e^{u^{3} + {u'}^{3}} {u'}^{3} u^{2} u' + 3 e^{u^{3} + {u'}^{3}} u^{2} u'$

Step 4

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

$1 = 3 e^{u^{3} + {u'}^{3}} u^{5} u' - 3 e^{u^{3} + {u'}^{3}} {u'}^{3} u^{2} u' + 3 e^{u^{3} + {u'}^{3}} u^{2} u'$

Step 5

Rewrite the equation as $3 e^{u^{3} + {u'}^{3}} u^{5} u' - 3 e^{u^{3} + {u'}^{3}} {u'}^{3} u^{2} u' + 3 e^{u^{3} + {u'}^{3}} u^{2} u' = 1$ .

$3 e^{u^{3} + {u'}^{3}} u^{5} u' - 3 e^{u^{3} + {u'}^{3}} {u'}^{3} u^{2} u' + 3 e^{u^{3} + {u'}^{3}} u^{2} u' = 1$

Step 6

Replace $u'$ with $\frac{d u}{d z}$ .

$\frac{d u}{d z} = 1$