Calculus Examples

$y = \sqrt{9 x} \cos (x)$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{9 x}$ as ${(9 x)}^{\frac{1}{2}}$ .

$y = {(9 x)}^{\frac{1}{2}} \cos (x)$

Step 2

Differentiate both sides of the equation.

$\frac{d}{d x} (y) = \frac{d}{d x} ({(9 x)}^{\frac{1}{2}} \cos (x))$

Step 3

The derivative of $y$ with respect to $x$ is $y'$ .

$y'$

Step 4

Differentiate the right side of the equation.

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Step 4.1

Factor $9$ out of $9 x$ .

$\frac{d}{d x} [{(9 (x))}^{\frac{1}{2}} \cos (x)]$

Step 4.2

Simplify the expression.

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Step 4.2.1

Apply the product rule to $9 (x)$ .

$\frac{d}{d x} [9^{\frac{1}{2}} x^{\frac{1}{2}} \cos (x)]$

Step 4.2.2

Rewrite $9$ as $3^{2}$ .

$\frac{d}{d x} [{(3^{2})}^{\frac{1}{2}} x^{\frac{1}{2}} \cos (x)]$

Step 4.2.3

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{d}{d x} [3^{2 (\frac{1}{2})} x^{\frac{1}{2}} \cos (x)]$

Step 4.3

Cancel the common factor of $2$ .

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Step 4.3.1

Cancel the common factor.

$\frac{d}{d x} [3^{2 (\frac{1}{2})} x^{\frac{1}{2}} \cos (x)]$

Step 4.3.2

Rewrite the expression.

$\frac{d}{d x} [3^{1} x^{\frac{1}{2}} \cos (x)]$

Step 4.4

Evaluate the exponent.

$\frac{d}{d x} [3 x^{\frac{1}{2}} \cos (x)]$

Step 4.5

Since $3$ is constant with respect to $x$ , the derivative of $3 x^{\frac{1}{2}} \cos (x)$ with respect to $x$ is $3 \frac{d}{d x} [x^{\frac{1}{2}} \cos (x)]$ .

$3 \frac{d}{d x} [x^{\frac{1}{2}} \cos (x)]$

Step 4.6

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = x^{\frac{1}{2}}$ and $g (x) = \cos (x)$ .

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

Step 4.7

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

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

Step 4.8

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

$3 (x^{\frac{1}{2}} (- \sin (x)) + \cos (x) (\frac{1}{2} x^{\frac{1}{2} - 1}))$

Step 4.9

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

$3 (x^{\frac{1}{2}} (- \sin (x)) + \cos (x) (\frac{1}{2} x^{\frac{1}{2} - 1 \cdot \frac{2}{2}}))$

Step 4.10

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

$3 (x^{\frac{1}{2}} (- \sin (x)) + \cos (x) (\frac{1}{2} x^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}}))$

Step 4.11

Combine the numerators over the common denominator.

$3 (x^{\frac{1}{2}} (- \sin (x)) + \cos (x) (\frac{1}{2} x^{\frac{1 - 1 \cdot 2}{2}}))$

Step 4.12

Simplify the numerator.

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Step 4.12.1

Multiply $- 1$ by $2$ .

$3 (x^{\frac{1}{2}} (- \sin (x)) + \cos (x) (\frac{1}{2} x^{\frac{1 - 2}{2}}))$

Step 4.12.2

Subtract $2$ from $1$ .

$3 (x^{\frac{1}{2}} (- \sin (x)) + \cos (x) (\frac{1}{2} x^{\frac{- 1}{2}}))$

Step 4.13

Move the negative in front of the fraction.

$3 (x^{\frac{1}{2}} (- \sin (x)) + \cos (x) (\frac{1}{2} x^{- \frac{1}{2}}))$

Step 4.14

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

$3 (x^{\frac{1}{2}} (- \sin (x)) + \cos (x) \frac{x^{- \frac{1}{2}}}{2})$

Step 4.15

Combine $\cos (x)$ and $\frac{x^{- \frac{1}{2}}}{2}$ .

$3 (x^{\frac{1}{2}} (- \sin (x)) + \frac{\cos (x) x^{- \frac{1}{2}}}{2})$

Step 4.16

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

$3 (x^{\frac{1}{2}} (- \sin (x)) + \frac{\cos (x)}{2 x^{\frac{1}{2}}})$

Step 4.17

Simplify.

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Step 4.17.1

Apply the distributive property.

$3 (x^{\frac{1}{2}} (- \sin (x))) + 3 \frac{\cos (x)}{2 x^{\frac{1}{2}}}$

Step 4.17.2

Combine terms.

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Step 4.17.2.1

Multiply $- 1$ by $3$ .

$- 3 (x^{\frac{1}{2}} (\sin (x))) + 3 \frac{\cos (x)}{2 x^{\frac{1}{2}}}$

Step 4.17.2.2

Combine $3$ and $\frac{\cos (x)}{2 x^{\frac{1}{2}}}$ .

$- 3 x^{\frac{1}{2}} \sin (x) + \frac{3 \cos (x)}{2 x^{\frac{1}{2}}}$

Step 5

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

$y' = - 3 x^{\frac{1}{2}} \sin (x) + \frac{3 \cos (x)}{2 x^{\frac{1}{2}}}$

Step 6

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

$\frac{d y}{d x} = - 3 x^{\frac{1}{2}} \sin (x) + \frac{3 \cos (x)}{2 x^{\frac{1}{2}}}$