Calculus Examples

$3 \sin (x) \cos (y) = 1$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (3 \sin (x) \cos (y)) = \frac{d}{d x} (1)$

Step 2

Differentiate the left side of the equation.

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Step 2.1

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

$3 \frac{d}{d x} [\sin (x) \cos (y)]$

Step 2.2

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 $f (x) = \sin (x)$ and $g (x) = \cos (y)$ .

$3 (\sin (x) \frac{d}{d x} [\cos (y)] + \cos (y) \frac{d}{d x} [\sin (x)])$

Step 2.3

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

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Step 2.3.1

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

$3 (\sin (x) (\frac{d}{d u} [\cos (u)] \frac{d}{d x} [y]) + \cos (y) \frac{d}{d x} [\sin (x)])$

Step 2.3.2

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

$3 (\sin (x) (- \sin (u) \frac{d}{d x} [y]) + \cos (y) \frac{d}{d x} [\sin (x)])$

Step 2.3.3

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

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

Step 2.4

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

$3 (\sin (x) (- \sin (y) y') + \cos (y) \frac{d}{d x} [\sin (x)])$

Step 2.5

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

$3 (\sin (x) (- \sin (y) y') + \cos (y) \cos (x))$

Step 2.6

Simplify.

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Step 2.6.1

Apply the distributive property.

$3 (\sin (x) (- \sin (y) y')) + 3 (\cos (y) \cos (x))$

Step 2.6.2

Multiply $- 1$ by $3$ .

$- 3 \sin (x) \sin (y) y' + 3 \cos (y) \cos (x)$

Step 2.6.3

Reorder terms.

$- 3 \sin (x) \sin (y) y' + 3 \cos (x) \cos (y)$

Step 3

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

$0$

Step 4

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

$- 3 \sin (x) \sin (y) y' + 3 \cos (x) \cos (y) = 0$

Step 5

Solve for $y'$ .

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Step 5.1

Simplify the left side.

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Step 5.1.1

Reorder factors in $- 3 \sin (x) \sin (y) y' + 3 \cos (x) \cos (y)$ .

$- 3 y' \sin (x) \sin (y) + 3 \cos (x) \cos (y) = 0$

Step 5.2

Subtract $3 \cos (x) \cos (y)$ from both sides of the equation.

$- 3 y' \sin (x) \sin (y) = - 3 \cos (x) \cos (y)$

Step 5.3

Divide each term in $- 3 y' \sin (x) \sin (y) = - 3 \cos (x) \cos (y)$ by $- 3 \sin (x) \sin (y)$ and simplify.

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Step 5.3.1

Divide each term in $- 3 y' \sin (x) \sin (y) = - 3 \cos (x) \cos (y)$ by $- 3 \sin (x) \sin (y)$ .

$\frac{- 3 y' \sin (x) \sin (y)}{- 3 \sin (x) \sin (y)} = \frac{- 3 \cos (x) \cos (y)}{- 3 \sin (x) \sin (y)}$

Step 5.3.2

Simplify the left side.

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Step 5.3.2.1

Cancel the common factor of $- 3$ .

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Step 5.3.2.1.1

Cancel the common factor.

$\frac{- 3 y' \sin (x) \sin (y)}{- 3 \sin (x) \sin (y)} = \frac{- 3 \cos (x) \cos (y)}{- 3 \sin (x) \sin (y)}$

Step 5.3.2.1.2

Rewrite the expression.

$\frac{y' \sin (x) \sin (y)}{\sin (x) \sin (y)} = \frac{- 3 \cos (x) \cos (y)}{- 3 \sin (x) \sin (y)}$

Step 5.3.2.2

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

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Step 5.3.2.2.1

Cancel the common factor.

$\frac{y' \sin (x) \sin (y)}{\sin (x) \sin (y)} = \frac{- 3 \cos (x) \cos (y)}{- 3 \sin (x) \sin (y)}$

Step 5.3.2.2.2

Rewrite the expression.

$\frac{y' \sin (y)}{\sin (y)} = \frac{- 3 \cos (x) \cos (y)}{- 3 \sin (x) \sin (y)}$

Step 5.3.2.3

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

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Step 5.3.2.3.1

Cancel the common factor.

$\frac{y' \sin (y)}{\sin (y)} = \frac{- 3 \cos (x) \cos (y)}{- 3 \sin (x) \sin (y)}$

Step 5.3.2.3.2

Divide $y'$ by $1$ .

$y' = \frac{- 3 \cos (x) \cos (y)}{- 3 \sin (x) \sin (y)}$

Step 5.3.3

Simplify the right side.

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Step 5.3.3.1

Cancel the common factor of $- 3$ .

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Step 5.3.3.1.1

Cancel the common factor.

$y' = \frac{- 3 \cos (x) \cos (y)}{- 3 \sin (x) \sin (y)}$

Step 5.3.3.1.2

Rewrite the expression.

$y' = \frac{\cos (x) \cos (y)}{\sin (x) \sin (y)}$

Step 5.3.3.2

Separate fractions.

$y' = \frac{\cos (y)}{\sin (y)} \cdot \frac{\cos (x)}{\sin (x)}$

Step 5.3.3.3

Convert from $\frac{\cos (x)}{\sin (x)}$ to $\cot (x)$ .

$y' = \frac{\cos (y)}{\sin (y)} \cot (x)$

Step 5.3.3.4

Convert from $\frac{\cos (y)}{\sin (y)}$ to $\cot (y)$ .

$y' = \cot (y) \cot (x)$

Step 6

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

$\frac{d y}{d x} = \cot (y) \cot (x)$