Calculus Examples

$\cos (x) + \sqrt{y} = 8$

Step 1

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

$\cos (x) + y^{\frac{1}{2}} = 8$

Step 2

Differentiate both sides of the equation.

$\frac{d}{d x} (\cos (x) + y^{\frac{1}{2}}) = \frac{d}{d x} (8)$

Step 3

Differentiate the left side of the equation.

Tap for more steps...

Step 3.1

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

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

Step 3.2

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

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

Step 3.3

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

Tap for more steps...

Step 3.3.1

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

Tap for more steps...

Step 3.3.1.1

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

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

Step 3.3.1.2

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

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

Step 3.3.1.3

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

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

Step 3.3.2

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

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

Step 3.3.3

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

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

Step 3.3.4

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

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

Step 3.3.5

Combine the numerators over the common denominator.

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

Step 3.3.6

Simplify the numerator.

Tap for more steps...

Step 3.3.6.1

Multiply $- 1$ by $2$ .

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

Step 3.3.6.2

Subtract $2$ from $1$ .

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

Step 3.3.7

Move the negative in front of the fraction.

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

Step 3.3.8

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

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

Step 3.3.9

Combine $\frac{y^{- \frac{1}{2}}}{2}$ and $y'$ .

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

Step 3.3.10

Move $y^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 3.4

Reorder terms.

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

Step 4

Since $8$ is constant with respect to $x$ , the derivative of $8$ with respect to $x$ is $0$ .

$0$

Step 5

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

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

Step 6

Solve for $y'$ .

Tap for more steps...

Step 6.1

Add $\sin (x)$ to both sides of the equation.

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

Step 6.2

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

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

Step 6.3

Simplify.

Tap for more steps...

Step 6.3.1

Simplify the left side.

Tap for more steps...

Step 6.3.1.1

Simplify $\frac{y'}{2 y^{\frac{1}{2}}} (2 y^{\frac{1}{2}})$ .

Tap for more steps...

Step 6.3.1.1.1

Rewrite using the commutative property of multiplication.

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

Step 6.3.1.1.2

Cancel the common factor of $2$ .

Tap for more steps...

Step 6.3.1.1.2.1

Cancel the common factor.

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

Step 6.3.1.1.2.2

Rewrite the expression.

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

Step 6.3.1.1.3

Cancel the common factor of $y^{\frac{1}{2}}$ .

Tap for more steps...

Step 6.3.1.1.3.1

Cancel the common factor.

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

Step 6.3.1.1.3.2

Rewrite the expression.

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

Step 6.3.2

Simplify the right side.

Tap for more steps...

Step 6.3.2.1

Simplify $\sin (x) (2 y^{\frac{1}{2}})$ .

Tap for more steps...

Step 6.3.2.1.1

Rewrite using the commutative property of multiplication.

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

Step 6.3.2.1.2

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

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

Step 7

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

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