Calculus Examples

$\cos (6 y) = x$ ; $(0, \frac{π}{12})$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d y} (\cos (6 y)) = \frac{d}{d y} (x)$

Step 2

Differentiate the left side of the equation.

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Step 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 $g (y) = 6 y$ .

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Step 2.1.1

To apply the Chain Rule, set $u$ as $6 y$ .

$\frac{d}{d u} [\cos (u)] \frac{d}{d y} [6 y]$

Step 2.1.2

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

$- \sin (u) \frac{d}{d y} [6 y]$

Step 2.1.3

Replace all occurrences of $u$ with $6 y$ .

$- \sin (6 y) \frac{d}{d y} [6 y]$

Step 2.2

Differentiate.

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Step 2.2.1

Since $6$ is constant with respect to $y$ , the derivative of $6 y$ with respect to $y$ is $6 \frac{d}{d y} [y]$ .

$- \sin (6 y) (6 \frac{d}{d y} [y])$

Step 2.2.2

Multiply $6$ by $- 1$ .

$- 6 \sin (6 y) \frac{d}{d y} [y]$

Step 2.2.3

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

$- 6 \sin (6 y) \cdot 1$

Step 2.2.4

Multiply $- 6$ by $1$ .

$- 6 \sin (6 y)$

Step 3

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

$x'$

Step 4

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

$- 6 \sin (6 y) = x'$

Step 5

Rewrite the equation as $x' = - 6 \sin (6 y)$ .

$x' = - 6 \sin (6 y)$

Step 6

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

$\frac{d x}{d y} = - 6 \sin (6 y)$

Step 7

Replace $x$ with $0$ and $y$ with $\frac{p}{12}$ in the expression.

$\frac{d x}{d y} = - 6 \sin (6 (\frac{p}{12}))$

Step 8

Cancel the common factor of $6$ .

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Step 8.1

Factor $6$ out of $12$ .

$- 6 \sin (6 \frac{p}{6 (2)})$

Step 8.2

Cancel the common factor.

$- 6 \sin (6 \frac{p}{6 \cdot 2})$

Step 8.3

Rewrite the expression.

$- 6 \sin (\frac{p}{2})$