Algebra Examples

$y = {(10 x^{3} + 7)}^{\frac{3}{2}}$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d y} (y) = \frac{d}{d y} ({(10 x^{3} + 7)}^{\frac{3}{2}})$

Step 2

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

$1$

Step 3

Differentiate the right side of the equation.

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Step 3.1

Differentiate using the chain rule, which states that $\frac{d}{d y} [f (g (y))]$ is $f' (g (y)) g' (y)$ where $f (y) = y^{\frac{3}{2}}$ and $g (y) = 10 x^{3} + 7$ .

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Step 3.1.1

To apply the Chain Rule, set $u_{1}$ as $10 x^{3} + 7$ .

$\frac{d}{d u_{1}} [{u_{1}}^{\frac{3}{2}}] \frac{d}{d y} [10 x^{3} + 7]$

Step 3.1.2

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

$\frac{3}{2} {u_{1}}^{\frac{3}{2} - 1} \frac{d}{d y} [10 x^{3} + 7]$

Step 3.1.3

Replace all occurrences of $u_{1}$ with $10 x^{3} + 7$ .

$\frac{3}{2} {(10 x^{3} + 7)}^{\frac{3}{2} - 1} \frac{d}{d y} [10 x^{3} + 7]$

Step 3.2

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{3}{2} {(10 x^{3} + 7)}^{\frac{3}{2} - 1 \cdot \frac{2}{2}} \frac{d}{d y} [10 x^{3} + 7]$

Step 3.3

Combine $- 1$ and $\frac{2}{2}$ .

$\frac{3}{2} {(10 x^{3} + 7)}^{\frac{3}{2} + \frac{- 1 \cdot 2}{2}} \frac{d}{d y} [10 x^{3} + 7]$

Step 3.4

Combine the numerators over the common denominator.

$\frac{3}{2} {(10 x^{3} + 7)}^{\frac{3 - 1 \cdot 2}{2}} \frac{d}{d y} [10 x^{3} + 7]$

Step 3.5

Simplify the numerator.

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Step 3.5.1

Multiply $- 1$ by $2$ .

$\frac{3}{2} {(10 x^{3} + 7)}^{\frac{3 - 2}{2}} \frac{d}{d y} [10 x^{3} + 7]$

Step 3.5.2

Subtract $2$ from $3$ .

$\frac{3}{2} {(10 x^{3} + 7)}^{\frac{1}{2}} \frac{d}{d y} [10 x^{3} + 7]$

Step 3.6

Differentiate.

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Step 3.6.1

Combine $\frac{3}{2}$ and ${(10 x^{3} + 7)}^{\frac{1}{2}}$ .

$\frac{3 {(10 x^{3} + 7)}^{\frac{1}{2}}}{2} \frac{d}{d y} [10 x^{3} + 7]$

Step 3.6.2

By the Sum Rule, the derivative of $10 x^{3} + 7$ with respect to $y$ is $\frac{d}{d y} [10 x^{3}] + \frac{d}{d y} [7]$ .

$\frac{3 {(10 x^{3} + 7)}^{\frac{1}{2}}}{2} (\frac{d}{d y} [10 x^{3}] + \frac{d}{d y} [7])$

Step 3.6.3

Since $10$ is constant with respect to $y$ , the derivative of $10 x^{3}$ with respect to $y$ is $10 \frac{d}{d y} [x^{3}]$ .

$\frac{3 {(10 x^{3} + 7)}^{\frac{1}{2}}}{2} (10 \frac{d}{d y} [x^{3}] + \frac{d}{d y} [7])$

Step 3.7

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

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Step 3.7.1

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

$\frac{3 {(10 x^{3} + 7)}^{\frac{1}{2}}}{2} (10 (\frac{d}{d u_{2}} [{u_{2}}^{3}] \frac{d}{d y} [x]) + \frac{d}{d y} [7])$

Step 3.7.2

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

$\frac{3 {(10 x^{3} + 7)}^{\frac{1}{2}}}{2} (10 (3 {u_{2}}^{2} \frac{d}{d y} [x]) + \frac{d}{d y} [7])$

Step 3.7.3

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

$\frac{3 {(10 x^{3} + 7)}^{\frac{1}{2}}}{2} (10 (3 x^{2} \frac{d}{d y} [x]) + \frac{d}{d y} [7])$

Step 3.8

Multiply $3$ by $10$ .

$\frac{3 {(10 x^{3} + 7)}^{\frac{1}{2}}}{2} (30 (x^{2} \frac{d}{d y} [x]) + \frac{d}{d y} [7])$

Step 3.9

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

$\frac{3 {(10 x^{3} + 7)}^{\frac{1}{2}}}{2} (30 x^{2} x' + \frac{d}{d y} [7])$

Step 3.10

Since $7$ is constant with respect to $y$ , the derivative of $7$ with respect to $y$ is $0$ .

$\frac{3 {(10 x^{3} + 7)}^{\frac{1}{2}}}{2} (30 x^{2} x' + 0)$

Step 3.11

Simplify terms.

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Step 3.11.1

Add $30 x^{2} x'$ and $0$ .

$\frac{3 {(10 x^{3} + 7)}^{\frac{1}{2}}}{2} (30 x^{2} x')$

Step 3.11.2

Combine $30$ and $\frac{3 {(10 x^{3} + 7)}^{\frac{1}{2}}}{2}$ .

$\frac{30 (3 {(10 x^{3} + 7)}^{\frac{1}{2}})}{2} (x^{2} x')$

Step 3.11.3

Multiply $3$ by $30$ .

$\frac{90 {(10 x^{3} + 7)}^{\frac{1}{2}}}{2} (x^{2} x')$

Step 3.11.4

Combine $x^{2}$ and $\frac{90 {(10 x^{3} + 7)}^{\frac{1}{2}}}{2}$ .

$\frac{x^{2} (90 {(10 x^{3} + 7)}^{\frac{1}{2}})}{2} x'$

Step 3.11.5

Combine $\frac{x^{2} (90 {(10 x^{3} + 7)}^{\frac{1}{2}})}{2}$ and $x'$ .

$\frac{x^{2} (90 {(10 x^{3} + 7)}^{\frac{1}{2}}) x'}{2}$

Step 3.11.6

Factor $2$ out of $x^{2} (90 {(10 x^{3} + 7)}^{\frac{1}{2}}) x'$ .

$\frac{2 (x^{2} (45 {(10 x^{3} + 7)}^{\frac{1}{2}}) x')}{2}$

Step 3.12

Cancel the common factors.

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Step 3.12.1

Factor $2$ out of $2$ .

$\frac{2 (x^{2} (45 {(10 x^{3} + 7)}^{\frac{1}{2}}) x')}{2 (1)}$

Step 3.12.2

Cancel the common factor.

$\frac{2 (x^{2} (45 {(10 x^{3} + 7)}^{\frac{1}{2}}) x')}{2 \cdot 1}$

Step 3.12.3

Rewrite the expression.

$\frac{x^{2} (45 {(10 x^{3} + 7)}^{\frac{1}{2}}) x'}{1}$

Step 3.12.4

Divide $x^{2} (45 {(10 x^{3} + 7)}^{\frac{1}{2}}) x'$ by $1$ .

$x^{2} (45 {(10 x^{3} + 7)}^{\frac{1}{2}}) x'$

Step 3.13

Reorder the factors of $x^{2} (45 {(10 x^{3} + 7)}^{\frac{1}{2}}) x'$ .

$45 x^{2} {(10 x^{3} + 7)}^{\frac{1}{2}} x'$

Step 4

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

$1 = 45 x^{2} {(10 x^{3} + 7)}^{\frac{1}{2}} x'$

Step 5

Solve for $x'$ .

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Step 5.1

Rewrite the equation as $45 x^{2} {(10 x^{3} + 7)}^{\frac{1}{2}} x' = 1$ .

$45 x^{2} {(10 x^{3} + 7)}^{\frac{1}{2}} x' = 1$

Step 5.2

Divide each term in $45 x^{2} {(10 x^{3} + 7)}^{\frac{1}{2}} x' = 1$ by $45 x^{2} {(10 x^{3} + 7)}^{\frac{1}{2}}$ and simplify.

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Step 5.2.1

Divide each term in $45 x^{2} {(10 x^{3} + 7)}^{\frac{1}{2}} x' = 1$ by $45 x^{2} {(10 x^{3} + 7)}^{\frac{1}{2}}$ .

$\frac{45 x^{2} {(10 x^{3} + 7)}^{\frac{1}{2}} x'}{45 x^{2} {(10 x^{3} + 7)}^{\frac{1}{2}}} = \frac{1}{45 x^{2} {(10 x^{3} + 7)}^{\frac{1}{2}}}$

Step 5.2.2

Simplify the left side.

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Step 5.2.2.1

Cancel the common factor.

$\frac{45 x^{2} {(10 x^{3} + 7)}^{\frac{1}{2}} x'}{45 x^{2} {(10 x^{3} + 7)}^{\frac{1}{2}}} = \frac{1}{45 x^{2} {(10 x^{3} + 7)}^{\frac{1}{2}}}$

Step 5.2.2.2

Rewrite the expression.

$\frac{x^{2} {(10 x^{3} + 7)}^{\frac{1}{2}} x'}{x^{2} {(10 x^{3} + 7)}^{\frac{1}{2}}} = \frac{1}{45 x^{2} {(10 x^{3} + 7)}^{\frac{1}{2}}}$

Step 5.2.2.3

Cancel the common factor.

$\frac{x^{2} {(10 x^{3} + 7)}^{\frac{1}{2}} x'}{x^{2} {(10 x^{3} + 7)}^{\frac{1}{2}}} = \frac{1}{45 x^{2} {(10 x^{3} + 7)}^{\frac{1}{2}}}$

Step 5.2.2.4

Rewrite the expression.

$\frac{{(10 x^{3} + 7)}^{\frac{1}{2}} x'}{{(10 x^{3} + 7)}^{\frac{1}{2}}} = \frac{1}{45 x^{2} {(10 x^{3} + 7)}^{\frac{1}{2}}}$

Step 5.2.2.5

Cancel the common factor.

$\frac{{(10 x^{3} + 7)}^{\frac{1}{2}} x'}{{(10 x^{3} + 7)}^{\frac{1}{2}}} = \frac{1}{45 x^{2} {(10 x^{3} + 7)}^{\frac{1}{2}}}$

Step 5.2.2.6

Divide $x'$ by $1$ .

$x' = \frac{1}{45 x^{2} {(10 x^{3} + 7)}^{\frac{1}{2}}}$

Step 6

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

$\frac{d x}{d y} = \frac{1}{45 x^{2} {(10 x^{3} + 7)}^{\frac{1}{2}}}$