Calculus Examples

$v = \frac{1}{3} \cdot (p (2 a x^{2} - x^{3}))$

Step 1

Remove parentheses.

$x' = \frac{1}{3} \cdot (p (2 a x^{2} - x^{3}))$

Step 2

Differentiate both sides of the equation.

$\frac{d}{d x'} x' = \frac{d}{d x'} (\frac{1}{3} \cdot (p (2 a x^{2} - x^{3})))$

Step 3

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

$1$

Step 4

Differentiate the right side of the equation.

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Step 4.1

Differentiate.

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Step 4.1.1

Combine $p$ and $\frac{1}{3}$ .

$\frac{d}{d v} [\frac{p}{3} \cdot (2 a x^{2} - x^{3})]$

Step 4.1.2

Since $\frac{p}{3}$ is constant with respect to $x'$ , the derivative of $\frac{p}{3} (2 a x^{2} - x^{3})$ with respect to $x'$ is $\frac{p}{3} \frac{d}{d v} [2 a x^{2} - x^{3}]$ .

$\frac{p}{3} \frac{d}{d v} [2 a x^{2} - x^{3}]$

Step 4.1.3

By the Sum Rule, the derivative of $2 a x^{2} - x^{3}$ with respect to $x'$ is $\frac{d}{d v} [2 a x^{2}] + \frac{d}{d v} [- x^{3}]$ .

$\frac{p}{3} (\frac{d}{d v} [2 a x^{2}] + \frac{d}{d v} [- x^{3}])$

Step 4.1.4

Since $2 a$ is constant with respect to $x'$ , the derivative of $2 a x^{2}$ with respect to $x'$ is $2 a \frac{d}{d v} [x^{2}]$ .

$\frac{p}{3} (2 a \frac{d}{d v} [x^{2}] + \frac{d}{d v} [- x^{3}])$

Step 4.2

Differentiate using the chain rule, which states that $\frac{d}{d v} [f (g (x'))]$ is $f' (g (x')) g' (x')$ where $f (x') = {x'}^{2}$ and $g (x') = x$ .

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Step 4.2.1

To apply the Chain Rule, set $u_{1}$ as $x$ .

$\frac{p}{3} (2 a (\frac{d}{d u_{1}} [{u_{1}}^{2}] \frac{d}{d v} [x]) + \frac{d}{d v} [- x^{3}])$

Step 4.2.2

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

$\frac{p}{3} (2 a (2 u_{1} \frac{d}{d v} [x]) + \frac{d}{d v} [- x^{3}])$

Step 4.2.3

Replace all occurrences of $u_{1}$ with $x$ .

$\frac{p}{3} (2 a (2 x \frac{d}{d v} [x]) + \frac{d}{d v} [- x^{3}])$

Step 4.3

Multiply $2$ by $2$ .

$\frac{p}{3} (4 a (x \frac{d}{d v} [x]) + \frac{d}{d v} [- x^{3}])$

Step 4.4

Rewrite $\frac{d}{d v} [x]$ as $x'$ .

$\frac{p}{3} (4 a x x' + \frac{d}{d v} [- x^{3}])$

Step 4.5

Since $- 1$ is constant with respect to $x'$ , the derivative of $- x^{3}$ with respect to $x'$ is $- \frac{d}{d v} [x^{3}]$ .

$\frac{p}{3} (4 a x x' - \frac{d}{d v} [x^{3}])$

Step 4.6

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

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Step 4.6.1

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

$\frac{p}{3} (4 a x x' - (\frac{d}{d u_{2}} [{u_{2}}^{3}] \frac{d}{d v} [x]))$

Step 4.6.2

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

$\frac{p}{3} (4 a x x' - (3 {u_{2}}^{2} \frac{d}{d v} [x]))$

Step 4.6.3

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

$\frac{p}{3} (4 a x x' - (3 x^{2} \frac{d}{d v} [x]))$

Step 4.7

Multiply $3$ by $- 1$ .

$\frac{p}{3} (4 a x x' - 3 (x^{2} \frac{d}{d v} [x]))$

Step 4.8

Rewrite $\frac{d}{d v} [x]$ as $x'$ .

$\frac{p}{3} (4 a x x' - 3 x^{2} x')$

Step 4.9

Simplify.

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Step 4.9.1

Apply the distributive property.

$\frac{p}{3} (4 a x x') + \frac{p}{3} (- 3 x^{2} x')$

Step 4.9.2

Combine terms.

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Step 4.9.2.1

Combine $4$ and $\frac{p}{3}$ .

$\frac{4 p}{3} (a x x') + \frac{p}{3} (- 3 x^{2} x')$

Step 4.9.2.2

Combine $a$ and $\frac{4 p}{3}$ .

$\frac{a (4 p)}{3} (x x') + \frac{p}{3} (- 3 x^{2} x')$

Step 4.9.2.3

Combine $x$ and $\frac{a (4 p)}{3}$ .

$\frac{x (a (4 p))}{3} x' + \frac{p}{3} (- 3 x^{2} x')$

Step 4.9.2.4

Combine $\frac{x (a (4 p))}{3}$ and $x'$ .

$\frac{x (a (4 p)) x'}{3} + \frac{p}{3} (- 3 x^{2} x')$

Step 4.9.2.5

Move $4$ to the left of $a$ .

$\frac{x (4 \cdot a p) x'}{3} + \frac{p}{3} (- 3 x^{2} x')$

Step 4.9.2.6

Move $4$ to the left of $x$ .

$\frac{4 \cdot x a p x'}{3} + \frac{p}{3} (- 3 x^{2} x')$

Step 4.9.2.7

Combine $- 3$ and $\frac{p}{3}$ .

$\frac{4 x a p x'}{3} + \frac{- 3 p}{3} (x^{2} x')$

Step 4.9.2.8

Combine $x^{2}$ and $\frac{- 3 p}{3}$ .

$\frac{4 x a p x'}{3} + \frac{x^{2} (- 3 p)}{3} x'$

Step 4.9.2.9

Combine $\frac{x^{2} (- 3 p)}{3}$ and $x'$ .

$\frac{4 x a p x'}{3} + \frac{x^{2} (- 3 p) x'}{3}$

Step 4.9.2.10

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

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Step 4.9.2.10.1

Factor $3$ out of $x^{2} (- 3 p) x'$ .

$\frac{4 x a p x'}{3} + \frac{3 (x^{2} (- p) x')}{3}$

Step 4.9.2.10.2

Cancel the common factors.

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Step 4.9.2.10.2.1

Factor $3$ out of $3$ .

$\frac{4 x a p x'}{3} + \frac{3 (x^{2} (- p) x')}{3 (1)}$

Step 4.9.2.10.2.2

Cancel the common factor.

$\frac{4 x a p x'}{3} + \frac{3 (x^{2} (- p) x')}{3 \cdot 1}$

Step 4.9.2.10.2.3

Rewrite the expression.

$\frac{4 x a p x'}{3} + \frac{x^{2} (- p) x'}{1}$

Step 4.9.2.10.2.4

Divide $x^{2} (- p) x'$ by $1$ .

$\frac{4 x a p x'}{3} + x^{2} (- p) x'$

Step 4.9.3

Reorder terms.

$\frac{4 x a p x'}{3} - x^{2} p x'$

Step 5

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

$1 = \frac{4 x a p x'}{3} - x^{2} p x'$

Step 6

Solve for $x'$ .

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Step 6.1

Rewrite the equation as $\frac{4 x a p x'}{3} - x^{2} p x' = 1$ .

$\frac{4 x a p x'}{3} - x^{2} p x' = 1$

Step 6.2

Factor $x p x'$ out of $\frac{4 x a p x'}{3} - x^{2} p x'$ .

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Step 6.2.1

Factor $x p x'$ out of $\frac{4 x a p x'}{3}$ .

$x p x' \frac{4 a}{3} - x^{2} p x' = 1$

Step 6.2.2

Factor $x p x'$ out of $- x^{2} p x'$ .

$x p x' \frac{4 a}{3} + x p x' (- x) = 1$

Step 6.2.3

Factor $x p x'$ out of $x p x' \frac{4 a}{3} + x p x' (- x)$ .

$x p x' (\frac{4 a}{3} - x) = 1$

Step 6.3

Divide each term in $x p x' (\frac{4 a}{3} - x) = 1$ by $x p (\frac{4 a}{3} - x)$ and simplify.

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Step 6.3.1

Divide each term in $x p x' (\frac{4 a}{3} - x) = 1$ by $x p (\frac{4 a}{3} - x)$ .

$\frac{x p x' (\frac{4 a}{3} - x)}{x p (\frac{4 a}{3} - x)} = \frac{1}{x p (\frac{4 a}{3} - x)}$

Step 6.3.2

Simplify the left side.

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Step 6.3.2.1

Cancel the common factor of $x$ .

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Step 6.3.2.1.1

Cancel the common factor.

$\frac{x p x' (\frac{4 a}{3} - x)}{x p (\frac{4 a}{3} - x)} = \frac{1}{x p (\frac{4 a}{3} - x)}$

Step 6.3.2.1.2

Rewrite the expression.

$\frac{p x' (\frac{4 a}{3} - x)}{p (\frac{4 a}{3} - x)} = \frac{1}{x p (\frac{4 a}{3} - x)}$

Step 6.3.2.2

Cancel the common factor of $p$ .

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Step 6.3.2.2.1

Cancel the common factor.

$\frac{p x' (\frac{4 a}{3} - x)}{p (\frac{4 a}{3} - x)} = \frac{1}{x p (\frac{4 a}{3} - x)}$

Step 6.3.2.2.2

Rewrite the expression.

$\frac{x' (\frac{4 a}{3} - x)}{\frac{4 a}{3} - x} = \frac{1}{x p (\frac{4 a}{3} - x)}$

Step 6.3.2.3

Cancel the common factor of $\frac{4 a}{3} - x$ .

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Step 6.3.2.3.1

Cancel the common factor.

$\frac{x' (\frac{4 a}{3} - x)}{\frac{4 a}{3} - x} = \frac{1}{x p (\frac{4 a}{3} - x)}$

Step 6.3.2.3.2

Divide $x'$ by $1$ .

$x' = \frac{1}{x p (\frac{4 a}{3} - x)}$

Step 6.3.3

Simplify the right side.

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Step 6.3.3.1

Simplify the denominator.

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Step 6.3.3.1.1

To write $- x$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$x' = \frac{1}{x p (\frac{4 a}{3} - x \cdot \frac{3}{3})}$

Step 6.3.3.1.2

Combine $- x$ and $\frac{3}{3}$ .

$x' = \frac{1}{x p (\frac{4 a}{3} + \frac{- x \cdot 3}{3})}$

Step 6.3.3.1.3

Combine the numerators over the common denominator.

$x' = \frac{1}{x p \frac{4 a - x \cdot 3}{3}}$

Step 6.3.3.1.4

Multiply $3$ by $- 1$ .

$x' = \frac{1}{x p \frac{4 a - 3 x}{3}}$

Step 6.3.3.1.5

Combine exponents.

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Step 6.3.3.1.5.1

Combine $x$ and $\frac{4 a - 3 x}{3}$ .

$x' = \frac{1}{p \frac{x (4 a - 3 x)}{3}}$

Step 6.3.3.1.5.2

Combine $p$ and $\frac{x (4 a - 3 x)}{3}$ .

$x' = \frac{1}{\frac{p (x (4 a - 3 x))}{3}}$

Step 6.3.3.1.6

Remove unnecessary parentheses.

$x' = \frac{1}{\frac{p x (4 a - 3 x)}{3}}$

Step 6.3.3.2

Multiply the numerator by the reciprocal of the denominator.

$x' = 1 \frac{3}{p x (4 a - 3 x)}$

Step 6.3.3.3

Multiply $\frac{3}{p x (4 a - 3 x)}$ by $1$ .

$x' = \frac{3}{p x (4 a - 3 x)}$

Step 7

Replace $x'$ with $\frac{d x}{d v}$ .

$\frac{d x}{d v} = \frac{3}{p x (4 a - 3 x)}$