Calculus Examples

$\cos (x) + \sqrt{y} = 6$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{y}$ as $y^{\frac{1}{2}}$ .

$\cos (x) + y^{\frac{1}{2}} = 6$

Step 2

Differentiate both sides of the equation.

$\frac{d}{d y} (\cos (x) + y^{\frac{1}{2}}) = \frac{d}{d y} (6)$

Step 3

Differentiate the left side of the equation.

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Step 3.1

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

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

Step 3.2

Evaluate $\frac{d}{d y} [\cos (x)]$ .

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Step 3.2.1

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

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Step 3.2.1.1

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

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

Step 3.2.1.2

The derivative of $\cos (u)$ with respect to $u$ is $- \sin (u)$ .

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

Step 3.2.1.3

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

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

Step 3.2.2

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

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

Step 3.3

Evaluate $\frac{d}{d y} [y^{\frac{1}{2}}]$ .

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Step 3.3.1

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

$- \sin (x) x' + \frac{1}{2} y^{\frac{1}{2} - 1}$

Step 3.3.2

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

$- \sin (x) x' + \frac{1}{2} y^{\frac{1}{2} - 1 \cdot \frac{2}{2}}$

Step 3.3.3

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

$- \sin (x) x' + \frac{1}{2} y^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}}$

Step 3.3.4

Combine the numerators over the common denominator.

$- \sin (x) x' + \frac{1}{2} y^{\frac{1 - 1 \cdot 2}{2}}$

Step 3.3.5

Simplify the numerator.

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Step 3.3.5.1

Multiply $- 1$ by $2$ .

$- \sin (x) x' + \frac{1}{2} y^{\frac{1 - 2}{2}}$

Step 3.3.5.2

Subtract $2$ from $1$ .

$- \sin (x) x' + \frac{1}{2} y^{\frac{- 1}{2}}$

Step 3.3.6

Move the negative in front of the fraction.

$- \sin (x) x' + \frac{1}{2} y^{- \frac{1}{2}}$

Step 3.4

Simplify.

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Step 3.4.1

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

$- \sin (x) x' + \frac{1}{2} \cdot \frac{1}{y^{\frac{1}{2}}}$

Step 3.4.2

Multiply $\frac{1}{2}$ by $\frac{1}{y^{\frac{1}{2}}}$ .

$- \sin (x) x' + \frac{1}{2 y^{\frac{1}{2}}}$

Step 4

Since $6$ is constant with respect to $y$ , the derivative of $6$ with respect to $y$ is $0$ .

$0$

Step 5

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

$- \sin (x) x' + \frac{1}{2 y^{\frac{1}{2}}} = 0$

Step 6

Solve for $x'$ .

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Step 6.1

Simplify the left side.

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Step 6.1.1

Reorder factors in $- \sin (x) x' + \frac{1}{2 y^{\frac{1}{2}}}$ .

$- x' \sin (x) + \frac{1}{2 y^{\frac{1}{2}}} = 0$

Step 6.2

Subtract $\frac{1}{2 y^{\frac{1}{2}}}$ from both sides of the equation.

$- x' \sin (x) = - \frac{1}{2 y^{\frac{1}{2}}}$

Step 6.3

Divide each term in $- x' \sin (x) = - \frac{1}{2 y^{\frac{1}{2}}}$ by $- \sin (x)$ and simplify.

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Step 6.3.1

Divide each term in $- x' \sin (x) = - \frac{1}{2 y^{\frac{1}{2}}}$ by $- \sin (x)$ .

$\frac{- x' \sin (x)}{- \sin (x)} = \frac{- \frac{1}{2 y^{\frac{1}{2}}}}{- \sin (x)}$

Step 6.3.2

Simplify the left side.

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Step 6.3.2.1

Dividing two negative values results in a positive value.

$\frac{x' \sin (x)}{\sin (x)} = \frac{- \frac{1}{2 y^{\frac{1}{2}}}}{- \sin (x)}$

Step 6.3.2.2

Cancel the common factor of $\sin (x)$ .

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Step 6.3.2.2.1

Cancel the common factor.

$\frac{x' \sin (x)}{\sin (x)} = \frac{- \frac{1}{2 y^{\frac{1}{2}}}}{- \sin (x)}$

Step 6.3.2.2.2

Divide $x'$ by $1$ .

$x' = \frac{- \frac{1}{2 y^{\frac{1}{2}}}}{- \sin (x)}$

Step 6.3.3

Simplify the right side.

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Step 6.3.3.1

Dividing two negative values results in a positive value.

$x' = \frac{\frac{1}{2 y^{\frac{1}{2}}}}{\sin (x)}$

Step 6.3.3.2

Separate fractions.

$x' = \frac{\frac{1}{2 y^{\frac{1}{2}}}}{1} \cdot \frac{1}{\sin (x)}$

Step 6.3.3.3

Convert from $\frac{1}{\sin (x)}$ to $\csc (x)$ .

$x' = \frac{\frac{1}{2 y^{\frac{1}{2}}}}{1} \csc (x)$

Step 6.3.3.4

Divide $\frac{1}{2 y^{\frac{1}{2}}}$ by $1$ .

$x' = \frac{1}{2 y^{\frac{1}{2}}} \csc (x)$

Step 6.3.3.5

Combine $\frac{1}{2 y^{\frac{1}{2}}}$ and $\csc (x)$ .

$x' = \frac{\csc (x)}{2 y^{\frac{1}{2}}}$

Step 7

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

$\frac{d x}{d y} = \frac{\csc (x)}{2 y^{\frac{1}{2}}}$