Calculus Examples

$- \frac{12 y^{\frac{3}{2}}}{1 + 12 x y^{\frac{1}{2}}}$

Step 1

This derivative could not be completed using the quotient rule. Mathway will use another method.

Step 2

Differentiate using the Constant Multiple Rule.

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Step 2.1

Since $- 12 y^{\frac{3}{2}}$ is constant with respect to $x$ , the derivative of $- \frac{12 y^{\frac{3}{2}}}{1 + 12 x y^{\frac{1}{2}}}$ with respect to $x$ is $- 12 y^{\frac{3}{2}} \frac{d}{d x} [\frac{1}{1 + 12 x y^{\frac{1}{2}}}]$ .

$- 12 y^{\frac{3}{2}} \frac{d}{d x} [\frac{1}{1 + 12 x y^{\frac{1}{2}}}]$

Step 2.2

Rewrite $\frac{1}{1 + 12 x y^{\frac{1}{2}}}$ as ${(1 + 12 x y^{\frac{1}{2}})}^{- 1}$ .

$- 12 y^{\frac{3}{2}} \frac{d}{d x} [{(1 + 12 x y^{\frac{1}{2}})}^{- 1}]$

Step 3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{- 1}$ and $g (x) = 1 + 12 x y^{\frac{1}{2}}$ .

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Step 3.1

To apply the Chain Rule, set $u$ as $1 + 12 x y^{\frac{1}{2}}$ .

$- 12 y^{\frac{3}{2}} (\frac{d}{d u} [u^{- 1}] \frac{d}{d x} [1 + 12 x y^{\frac{1}{2}}])$

Step 3.2

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

$- 12 y^{\frac{3}{2}} (- u^{- 2} \frac{d}{d x} [1 + 12 x y^{\frac{1}{2}}])$

Step 3.3

Replace all occurrences of $u$ with $1 + 12 x y^{\frac{1}{2}}$ .

$- 12 y^{\frac{3}{2}} (- {(1 + 12 x y^{\frac{1}{2}})}^{- 2} \frac{d}{d x} [1 + 12 x y^{\frac{1}{2}}])$

Step 4

Differentiate.

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Step 4.1

Multiply $- 1$ by $- 12$ .

$12 y^{\frac{3}{2}} ({(1 + 12 x y^{\frac{1}{2}})}^{- 2} \frac{d}{d x} [1 + 12 x y^{\frac{1}{2}}])$

Step 4.2

By the Sum Rule, the derivative of $1 + 12 x y^{\frac{1}{2}}$ with respect to $x$ is $\frac{d}{d x} [1] + \frac{d}{d x} [12 x y^{\frac{1}{2}}]$ .

$12 y^{\frac{3}{2}} ({(1 + 12 x y^{\frac{1}{2}})}^{- 2} (\frac{d}{d x} [1] + \frac{d}{d x} [12 x y^{\frac{1}{2}}]))$

Step 4.3

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$12 y^{\frac{3}{2}} ({(1 + 12 x y^{\frac{1}{2}})}^{- 2} (0 + \frac{d}{d x} [12 x y^{\frac{1}{2}}]))$

Step 4.4

Add $0$ and $\frac{d}{d x} [12 x y^{\frac{1}{2}}]$ .

$12 y^{\frac{3}{2}} ({(1 + 12 x y^{\frac{1}{2}})}^{- 2} \frac{d}{d x} [12 x y^{\frac{1}{2}}])$

Step 4.5

Since $12 y^{\frac{1}{2}}$ is constant with respect to $x$ , the derivative of $12 x y^{\frac{1}{2}}$ with respect to $x$ is $12 y^{\frac{1}{2}} \frac{d}{d x} [x]$ .

$12 y^{\frac{3}{2}} ({(1 + 12 x y^{\frac{1}{2}})}^{- 2} (12 y^{\frac{1}{2}} \frac{d}{d x} [x]))$

Step 4.6

Simplify the expression.

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Step 4.6.1

Move $12$ to the left of ${(1 + 12 x y^{\frac{1}{2}})}^{- 2}$ .

$12 y^{\frac{3}{2}} (12 \cdot {(1 + 12 x y^{\frac{1}{2}})}^{- 2} y^{\frac{1}{2}} \frac{d}{d x} [x])$

Step 4.6.2

Multiply $12$ by $12$ .

$144 y^{\frac{3}{2}} ({(1 + 12 x y^{\frac{1}{2}})}^{- 2} y^{\frac{1}{2}} \frac{d}{d x} [x])$

Step 5

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

$144 y^{\frac{1}{2} + \frac{3}{2}} ({(1 + 12 x y^{\frac{1}{2}})}^{- 2} \frac{d}{d x} [x])$

Step 6

Reduce the expression by cancelling the common factors.

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Step 6.1

Combine the numerators over the common denominator.

$144 y^{\frac{1 + 3}{2}} ({(1 + 12 x y^{\frac{1}{2}})}^{- 2} \frac{d}{d x} [x])$

Step 6.2

Add $1$ and $3$ .

$144 y^{\frac{4}{2}} ({(1 + 12 x y^{\frac{1}{2}})}^{- 2} \frac{d}{d x} [x])$

Step 6.3

Cancel the common factor of $4$ and $2$ .

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Step 6.3.1

Factor $2$ out of $4$ .

$144 y^{\frac{2 \cdot 2}{2}} ({(1 + 12 x y^{\frac{1}{2}})}^{- 2} \frac{d}{d x} [x])$

Step 6.3.2

Cancel the common factors.

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Step 6.3.2.1

Factor $2$ out of $2$ .

$144 y^{\frac{2 \cdot 2}{2 (1)}} ({(1 + 12 x y^{\frac{1}{2}})}^{- 2} \frac{d}{d x} [x])$

Step 6.3.2.2

Cancel the common factor.

$144 y^{\frac{2 \cdot 2}{2 \cdot 1}} ({(1 + 12 x y^{\frac{1}{2}})}^{- 2} \frac{d}{d x} [x])$

Step 6.3.2.3

Rewrite the expression.

$144 y^{\frac{2}{1}} ({(1 + 12 x y^{\frac{1}{2}})}^{- 2} \frac{d}{d x} [x])$

Step 6.3.2.4

Divide $2$ by $1$ .

$144 y^{2} ({(1 + 12 x y^{\frac{1}{2}})}^{- 2} \frac{d}{d x} [x])$

Step 7

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

$144 y^{2} ({(1 + 12 x y^{\frac{1}{2}})}^{- 2} \cdot 1)$

Step 8

Multiply ${(1 + 12 x y^{\frac{1}{2}})}^{- 2}$ by $1$ .

$144 y^{2} {(1 + 12 x y^{\frac{1}{2}})}^{- 2}$

Step 9

Simplify.

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Step 9.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$144 y^{2} \frac{1}{{(1 + 12 x y^{\frac{1}{2}})}^{2}}$

Step 9.2

Combine terms.

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Step 9.2.1

Combine $144$ and $\frac{1}{{(1 + 12 x y^{\frac{1}{2}})}^{2}}$ .

$y^{2} \frac{144}{{(1 + 12 x y^{\frac{1}{2}})}^{2}}$

Step 9.2.2

Combine $y^{2}$ and $\frac{144}{{(1 + 12 x y^{\frac{1}{2}})}^{2}}$ .

$\frac{y^{2} \cdot 144}{{(1 + 12 x y^{\frac{1}{2}})}^{2}}$

Step 9.2.3

Move $144$ to the left of $y^{2}$ .

$\frac{144 y^{2}}{{(1 + 12 x y^{\frac{1}{2}})}^{2}}$

Step 9.3

Reorder terms.

$\frac{144 y^{2}}{{(12 y^{\frac{1}{2}} x + 1)}^{2}}$

Step 9.4

Reorder factors in $\frac{144 y^{2}}{{(12 y^{\frac{1}{2}} x + 1)}^{2}}$ .

$\frac{144 y^{2}}{{(12 x y^{\frac{1}{2}} + 1)}^{2}}$