Find da/db a^3+b^3=133

$a^{3} + b^{3} = 133$

Differentiate both sides of the equation.

$\frac{d}{d b} (a^{3} + b^{3}) = \frac{d}{d b} (133)$

Differentiate the left side of the equation.

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

$\frac{d}{d b} [a^{3}] + \frac{d}{d b} [b^{3}]$

Evaluate $\frac{d}{d b} [a^{3}]$ .

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

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

$\frac{d}{d u} [u^{3}] \frac{d}{d b} [a] + \frac{d}{d b} [b^{3}]$

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

$3 u^{2} \frac{d}{d b} [a] + \frac{d}{d b} [b^{3}]$

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

$3 a^{2} \frac{d}{d b} [a] + \frac{d}{d b} [b^{3}]$

Rewrite $\frac{d}{d b} [a]$ as $\frac{d}{d b} [a]$ .

$3 a^{2} \frac{d}{d b} [a] + \frac{d}{d b} [b^{3}]$

Differentiate using the Power Rule.

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

$3 a^{2} \frac{d}{d b} [a] + 3 b^{2}$

Reorder terms.

$3 b^{2} + 3 a^{2} \frac{d}{d b} [a]$

Since $133$ is constant with respect to $b$ , the derivative of $133$ with respect to $b$ is $0$ .

$0$

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

$3 b^{2} + 3 a^{2} a' = 0$

Solve for $a'$ .

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Subtract $3 b^{2}$ from both sides of the equation.

$3 a^{2} a' = - 3 b^{2}$

Divide each term by $3 a^{2}$ and simplify.

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Divide each term in $3 a^{2} a' = - 3 b^{2}$ by $3 a^{2}$ .

$\frac{3 a^{2} a'}{3 a^{2}} = \frac{- 3 b^{2}}{3 a^{2}}$

Simplify $\frac{3 a^{2} a'}{3 a^{2}}$ .

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Cancel the common factor of $3$ .

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Cancel the common factor.

$\frac{3 a^{2} a'}{3 a^{2}} = \frac{- 3 b^{2}}{3 a^{2}}$

Rewrite the expression.

$\frac{a^{2} a'}{a^{2}} = \frac{- 3 b^{2}}{3 a^{2}}$

Cancel the common factor of $a^{2}$ .

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Cancel the common factor.

$\frac{a^{2} a'}{a^{2}} = \frac{- 3 b^{2}}{3 a^{2}}$

Divide $a'$ by $1$ .

$a' = \frac{- 3 b^{2}}{3 a^{2}}$

Simplify $\frac{- 3 b^{2}}{3 a^{2}}$ .

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Cancel the common factor of $- 3$ and $3$ .

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Factor $3$ out of $- 3 b^{2}$ .

$a' = \frac{3 (- b^{2})}{3 a^{2}}$

Cancel the common factors.

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Factor $3$ out of $3 a^{2}$ .

$a' = \frac{3 (- b^{2})}{3 (a^{2})}$

Cancel the common factor.

$a' = \frac{3 (- b^{2})}{3 a^{2}}$

Rewrite the expression.

$a' = \frac{- b^{2}}{a^{2}}$

Move the negative in front of the fraction.

$a' = - \frac{b^{2}}{a^{2}}$

Replace $a'$ with $\frac{d a}{d b}$ .

$\frac{d a}{d b} = - \frac{b^{2}}{a^{2}}$

Find da/db a^3+b^3=133

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