Calculus Examples

$x \sin (3 x + 5 y) = y \cos (x)$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d y} (x \sin (3 x + 5 y)) = \frac{d}{d y} (y \cos (x))$

Step 2

Differentiate the left side of the equation.

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Step 2.1

Differentiate using the Product Rule which states that $\frac{d}{d y} [f (y) g (y)]$ is $f (y) \frac{d}{d y} [g (y)] + g (y) \frac{d}{d y} [f (y)]$ where $f (y) = x$ and $g (y) = \sin (3 x + 5 y)$ .

$x \frac{d}{d y} [\sin (3 x + 5 y)] + \sin (3 x + 5 y) \frac{d}{d y} [x]$

Step 2.2

Differentiate using the chain rule, which states that $\frac{d}{d y} [f (g (y))]$ is $f' (g (y)) g' (y)$ where $f (y) = \sin (y)$ and $g (y) = 3 x + 5 y$ .

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Step 2.2.1

To apply the Chain Rule, set $u$ as $3 x + 5 y$ .

$x (\frac{d}{d u} [\sin (u)] \frac{d}{d y} [3 x + 5 y]) + \sin (3 x + 5 y) \frac{d}{d y} [x]$

Step 2.2.2

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

$x (\cos (u) \frac{d}{d y} [3 x + 5 y]) + \sin (3 x + 5 y) \frac{d}{d y} [x]$

Step 2.2.3

Replace all occurrences of $u$ with $3 x + 5 y$ .

$x (\cos (3 x + 5 y) \frac{d}{d y} [3 x + 5 y]) + \sin (3 x + 5 y) \frac{d}{d y} [x]$

Step 2.3

Differentiate.

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Step 2.3.1

By the Sum Rule, the derivative of $3 x + 5 y$ with respect to $y$ is $\frac{d}{d y} [3 x] + \frac{d}{d y} [5 y]$ .

$x (\cos (3 x + 5 y) (\frac{d}{d y} [3 x] + \frac{d}{d y} [5 y])) + \sin (3 x + 5 y) \frac{d}{d y} [x]$

Step 2.3.2

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

$x (\cos (3 x + 5 y) (3 \frac{d}{d y} [x] + \frac{d}{d y} [5 y])) + \sin (3 x + 5 y) \frac{d}{d y} [x]$

Step 2.4

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

$x (\cos (3 x + 5 y) (3 x' + \frac{d}{d y} [5 y])) + \sin (3 x + 5 y) \frac{d}{d y} [x]$

Step 2.5

Since $5$ is constant with respect to $y$ , the derivative of $5 y$ with respect to $y$ is $5 \frac{d}{d y} [y]$ .

$x (\cos (3 x + 5 y) (3 x' + 5 \frac{d}{d y} [y])) + \sin (3 x + 5 y) \frac{d}{d y} [x]$

Step 2.6

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

$x (\cos (3 x + 5 y) (3 x' + 5 \cdot 1)) + \sin (3 x + 5 y) \frac{d}{d y} [x]$

Step 2.7

Multiply $5$ by $1$ .

$x (\cos (3 x + 5 y) (3 x' + 5)) + \sin (3 x + 5 y) \frac{d}{d y} [x]$

Step 2.8

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

$x (\cos (3 x + 5 y) (3 x' + 5)) + \sin (3 x + 5 y) x'$

Step 2.9

Reorder terms.

$x \cos (3 x + 5 y) (3 x' + 5) + \sin (3 x + 5 y) x'$

Step 3

Differentiate the right side of the equation.

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Step 3.1

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

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

Step 3.2

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

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

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

Step 3.2.2

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

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

Step 3.2.3

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

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

Step 3.3

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

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

Step 3.4

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

$y (- \sin (x) x') + \cos (x) \cdot 1$

Step 3.5

Simplify the expression.

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Step 3.5.1

Multiply $\cos (x)$ by $1$ .

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

Step 3.5.2

Reorder terms.

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

Step 4

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

$x \cos (3 x + 5 y) (3 x' + 5) + \sin (3 x + 5 y) x' = - y \sin (x) x' + \cos (x)$

Step 5

Solve for $x'$ .

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Step 5.1

Simplify the left side.

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Step 5.1.1

Simplify $x \cos (3 x + 5 y) (3 x' + 5) + \sin (3 x + 5 y) x'$ .

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Step 5.1.1.1

Simplify each term.

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Step 5.1.1.1.1

Apply the distributive property.

$x \cos (3 x + 5 y) (3 x') + x \cos (3 x + 5 y) \cdot 5 + \sin (3 x + 5 y) x' = - y \sin (x) x' + \cos (x)$

Step 5.1.1.1.2

Rewrite using the commutative property of multiplication.

$3 x \cos (3 x + 5 y) x' + x \cos (3 x + 5 y) \cdot 5 + \sin (3 x + 5 y) x' = - y \sin (x) x' + \cos (x)$

Step 5.1.1.1.3

Move $5$ to the left of $x \cos (3 x + 5 y)$ .

$3 x \cos (3 x + 5 y) x' + 5 \cdot (x \cos (3 x + 5 y)) + \sin (3 x + 5 y) x' = - y \sin (x) x' + \cos (x)$

Step 5.1.1.1.4

Remove parentheses.

$3 x \cos (3 x + 5 y) x' + 5 x \cos (3 x + 5 y) + \sin (3 x + 5 y) x' = - y \sin (x) x' + \cos (x)$

Step 5.1.1.2

Reorder factors in $3 x \cos (3 x + 5 y) x' + 5 x \cos (3 x + 5 y) + \sin (3 x + 5 y) x'$ .

$3 x x' \cos (3 x + 5 y) + 5 x \cos (3 x + 5 y) + x' \sin (3 x + 5 y) = - y \sin (x) x' + \cos (x)$

Step 5.2

Simplify the right side.

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Step 5.2.1

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

$3 x x' \cos (3 x + 5 y) + 5 x \cos (3 x + 5 y) + x' \sin (3 x + 5 y) = - y x' \sin (x) + \cos (x)$

Step 5.3

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

$3 x x' \cos (3 x + 5 y) + 5 x \cos (3 x + 5 y) + x' \sin (3 x + 5 y) + y x' \sin (x) = \cos (x)$

Step 5.4

Subtract $5 x \cos (3 x + 5 y)$ from both sides of the equation.

$3 x x' \cos (3 x + 5 y) + x' \sin (3 x + 5 y) + y x' \sin (x) = \cos (x) - 5 x \cos (3 x + 5 y)$

Step 5.5

Factor $x'$ out of $3 x x' \cos (3 x + 5 y) + x' \sin (3 x + 5 y) + y x' \sin (x)$ .

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Step 5.5.1

Factor $x'$ out of $3 x x' \cos (3 x + 5 y)$ .

$x' (3 x \cos (3 x + 5 y)) + x' (\sin (3 x + 5 y)) + y (x') \sin (x) = \cos (x) - 5 x \cos (3 x + 5 y)$

Step 5.5.2

Factor $x'$ out of $x' \sin (3 x + 5 y)$ .

$x' (3 x \cos (3 x + 5 y)) + x' (\sin (3 x + 5 y)) + y (x') \sin (x) = \cos (x) - 5 x \cos (3 x + 5 y)$

Step 5.5.3

Factor $x'$ out of $y x' \sin (x)$ .

$x' (3 x \cos (3 x + 5 y)) + x' (\sin (3 x + 5 y)) + x' (y \sin (x)) = \cos (x) - 5 x \cos (3 x + 5 y)$

Step 5.5.4

Factor $x'$ out of $x' (3 x \cos (3 x + 5 y)) + x' (\sin (3 x + 5 y))$ .

$x' (3 x \cos (3 x + 5 y) + \sin (3 x + 5 y)) + x' (y \sin (x)) = \cos (x) - 5 x \cos (3 x + 5 y)$

Step 5.5.5

Factor $x'$ out of $x' (3 x \cos (3 x + 5 y) + \sin (3 x + 5 y)) + x' (y \sin (x))$ .

$x' (3 x \cos (3 x + 5 y) + \sin (3 x + 5 y) + y \sin (x)) = \cos (x) - 5 x \cos (3 x + 5 y)$

Step 5.6

Divide each term in $x' (3 x \cos (3 x + 5 y) + \sin (3 x + 5 y) + y \sin (x)) = \cos (x) - 5 x \cos (3 x + 5 y)$ by $3 x \cos (3 x + 5 y) + \sin (3 x + 5 y) + y \sin (x)$ and simplify.

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Step 5.6.1

Divide each term in $x' (3 x \cos (3 x + 5 y) + \sin (3 x + 5 y) + y \sin (x)) = \cos (x) - 5 x \cos (3 x + 5 y)$ by $3 x \cos (3 x + 5 y) + \sin (3 x + 5 y) + y \sin (x)$ .

$\frac{x' (3 x \cos (3 x + 5 y) + \sin (3 x + 5 y) + y \sin (x))}{3 x \cos (3 x + 5 y) + \sin (3 x + 5 y) + y \sin (x)} = \frac{\cos (x)}{3 x \cos (3 x + 5 y) + \sin (3 x + 5 y) + y \sin (x)} + \frac{- 5 x \cos (3 x + 5 y)}{3 x \cos (3 x + 5 y) + \sin (3 x + 5 y) + y \sin (x)}$

Step 5.6.2

Simplify the left side.

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Step 5.6.2.1

Cancel the common factor of $3 x \cos (3 x + 5 y) + \sin (3 x + 5 y) + y \sin (x)$ .

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Step 5.6.2.1.1

Cancel the common factor.

Step 5.6.2.1.2

Divide $x'$ by $1$ .

$x' = \frac{\cos (x)}{3 x \cos (3 x + 5 y) + \sin (3 x + 5 y) + y \sin (x)} + \frac{- 5 x \cos (3 x + 5 y)}{3 x \cos (3 x + 5 y) + \sin (3 x + 5 y) + y \sin (x)}$

Step 5.6.3

Simplify the right side.

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Step 5.6.3.1

Move the negative in front of the fraction.

$x' = \frac{\cos (x)}{3 x \cos (3 x + 5 y) + \sin (3 x + 5 y) + y \sin (x)} - \frac{5 x \cos (3 x + 5 y)}{3 x \cos (3 x + 5 y) + \sin (3 x + 5 y) + y \sin (x)}$

Step 5.6.3.2

Combine the numerators over the common denominator.

$x' = \frac{\cos (x) - 5 x \cos (3 x + 5 y)}{3 x \cos (3 x + 5 y) + \sin (3 x + 5 y) + y \sin (x)}$

Step 6

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

$\frac{d x}{d y} = \frac{\cos (x) - 5 x \cos (3 x + 5 y)}{3 x \cos (3 x + 5 y) + \sin (3 x + 5 y) + y \sin (x)}$