Calculus Examples

$\cot (y) = 4 x - 5 y$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (\cot (y)) = \frac{d}{d x} (4 x - 5 y)$

Step 2

Differentiate the left side of the equation.

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Step 2.1

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

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Step 2.1.1

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

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

Step 2.1.2

The derivative of $\cot (u)$ with respect to $u$ is $- \csc^{2} (u)$ .

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

Step 2.1.3

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

$- \csc^{2} (y) \frac{d}{d x} [y]$

Step 2.2

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

$- \csc^{2} (y) y'$

Step 3

Differentiate the right side of the equation.

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Step 3.1

By the Sum Rule, the derivative of $4 x - 5 y$ with respect to $x$ is $\frac{d}{d x} [4 x] + \frac{d}{d x} [- 5 y]$ .

$\frac{d}{d x} [4 x] + \frac{d}{d x} [- 5 y]$

Step 3.2

Evaluate $\frac{d}{d x} [4 x]$ .

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Step 3.2.1

Since $4$ is constant with respect to $x$ , the derivative of $4 x$ with respect to $x$ is $4 \frac{d}{d x} [x]$ .

$4 \frac{d}{d x} [x] + \frac{d}{d x} [- 5 y]$

Step 3.2.2

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

$4 \cdot 1 + \frac{d}{d x} [- 5 y]$

Step 3.2.3

Multiply $4$ by $1$ .

$4 + \frac{d}{d x} [- 5 y]$

Step 3.3

Evaluate $\frac{d}{d x} [- 5 y]$ .

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Step 3.3.1

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

$4 - 5 \frac{d}{d x} [y]$

Step 3.3.2

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

$4 - 5 y'$

Step 4

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

$- \csc^{2} (y) y' = 4 - 5 y'$

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 $- \csc^{2} (y) y'$ .

$- y' \csc^{2} (y) = 4 - 5 y'$

Step 5.2

Add $5 y'$ to both sides of the equation.

$- y' \csc^{2} (y) + 5 y' = 4$

Step 5.3

Factor $y'$ out of $- y' \csc^{2} (y) + 5 y'$ .

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Step 5.3.1

Factor $y'$ out of $- y' \csc^{2} (y)$ .

$y' (- 1 \csc^{2} (y)) + 5 y' = 4$

Step 5.3.2

Factor $y'$ out of $5 y'$ .

$y' (- 1 \csc^{2} (y)) + y' \cdot 5 = 4$

Step 5.3.3

Factor $y'$ out of $y' (- 1 \csc^{2} (y)) + y' \cdot 5$ .

$y' (- 1 \csc^{2} (y) + 5) = 4$

Step 5.4

Rewrite $- 1 \csc^{2} (y)$ as $- \csc^{2} (y)$ .

$y' (- \csc^{2} (y) + 5) = 4$

Step 5.5

Divide each term in $y' (- \csc^{2} (y) + 5) = 4$ by $- \csc^{2} (y) + 5$ and simplify.

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Step 5.5.1

Divide each term in $y' (- \csc^{2} (y) + 5) = 4$ by $- \csc^{2} (y) + 5$ .

$\frac{y' (- \csc^{2} (y) + 5)}{- \csc^{2} (y) + 5} = \frac{4}{- \csc^{2} (y) + 5}$

Step 5.5.2

Simplify the left side.

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Step 5.5.2.1

Cancel the common factor of $- \csc^{2} (y) + 5$ .

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Step 5.5.2.1.1

Cancel the common factor.

$\frac{y' (- \csc^{2} (y) + 5)}{- \csc^{2} (y) + 5} = \frac{4}{- \csc^{2} (y) + 5}$

Step 5.5.2.1.2

Divide $y'$ by $1$ .

$y' = \frac{4}{- \csc^{2} (y) + 5}$

Step 5.5.3

Simplify the right side.

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Step 5.5.3.1

Factor $- 1$ out of $- \csc^{2} (y)$ .

$y' = \frac{4}{- (\csc^{2} (y)) + 5}$

Step 5.5.3.2

Rewrite $5$ as $- 1 (- 5)$ .

$y' = \frac{4}{- (\csc^{2} (y)) - 1 (- 5)}$

Step 5.5.3.3

Factor $- 1$ out of $- (\csc^{2} (y)) - 1 (- 5)$ .

$y' = \frac{4}{- (\csc^{2} (y) - 5)}$

Step 5.5.3.4

Rewrite negatives.

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Step 5.5.3.4.1

Rewrite $- (\csc^{2} (y) - 5)$ as $- 1 (\csc^{2} (y) - 5)$ .

$y' = \frac{4}{- 1 (\csc^{2} (y) - 5)}$

Step 5.5.3.4.2

Move the negative in front of the fraction.

$y' = - \frac{4}{\csc^{2} (y) - 5}$

Step 6

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

$\frac{d y}{d x} = - \frac{4}{\csc^{2} (y) - 5}$