Calculus Examples

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

Step 1

Differentiate both sides of the equation.

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

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 $f (x) = \cos (x)$ and $g (x) = 5 x + y$ .

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Step 2.1.1

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

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

Step 2.1.2

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

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

Step 2.1.3

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

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

Step 2.2

Differentiate.

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Step 2.2.1

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

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

Step 2.2.2

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

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

Step 2.2.3

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

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

Step 2.2.4

Multiply $5$ by $1$ .

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

Step 2.3

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

$- \sin (5 x + y) (5 + y')$

Step 3

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

$y'$

Step 4

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

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

Step 5

Solve for $y'$ .

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Step 5.1

Simplify the left side.

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Step 5.1.1

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

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Step 5.1.1.1

Rewrite.

$0 + 0 - \sin (5 x + y) (5 + y') = y'$

Step 5.1.1.2

Simplify by adding zeros.

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

Step 5.1.1.3

Apply the distributive property.

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

Step 5.1.1.4

Simplify the expression.

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Step 5.1.1.4.1

Multiply $5$ by $- 1$ .

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

Step 5.1.1.4.2

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

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

Step 5.2

Subtract $y'$ from both sides of the equation.

$- 5 \sin (5 x + y) - y' \sin (5 x + y) - y' = 0$

Step 5.3

Add $5 \sin (5 x + y)$ to both sides of the equation.

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

Step 5.4

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

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Step 5.4.1

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

$y' (- 1 \sin (5 x + y)) - y' = 5 \sin (5 x + y)$

Step 5.4.2

Factor $y'$ out of $- y'$ .

$y' (- 1 \sin (5 x + y)) + y' \cdot - 1 = 5 \sin (5 x + y)$

Step 5.4.3

Factor $y'$ out of $y' (- 1 \sin (5 x + y)) + y' \cdot - 1$ .

$y' (- 1 \sin (5 x + y) - 1) = 5 \sin (5 x + y)$

Step 5.5

Rewrite $- 1 \sin (5 x + y)$ as $- \sin (5 x + y)$ .

$y' (- \sin (5 x + y) - 1) = 5 \sin (5 x + y)$

Step 5.6

Divide each term in $y' (- \sin (5 x + y) - 1) = 5 \sin (5 x + y)$ by $- \sin (5 x + y) - 1$ and simplify.

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Step 5.6.1

Divide each term in $y' (- \sin (5 x + y) - 1) = 5 \sin (5 x + y)$ by $- \sin (5 x + y) - 1$ .

$\frac{y' (- \sin (5 x + y) - 1)}{- \sin (5 x + y) - 1} = \frac{5 \sin (5 x + y)}{- \sin (5 x + y) - 1}$

Step 5.6.2

Simplify the left side.

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Step 5.6.2.1

Cancel the common factor of $- \sin (5 x + y) - 1$ .

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Step 5.6.2.1.1

Cancel the common factor.

$\frac{y' (- \sin (5 x + y) - 1)}{- \sin (5 x + y) - 1} = \frac{5 \sin (5 x + y)}{- \sin (5 x + y) - 1}$

Step 5.6.2.1.2

Divide $y'$ by $1$ .

$y' = \frac{5 \sin (5 x + y)}{- \sin (5 x + y) - 1}$

Step 5.6.3

Simplify the right side.

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Step 5.6.3.1

Factor $- 1$ out of $- \sin (5 x + y) - 1$ .

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Step 5.6.3.1.1

Factor $- 1$ out of $- \sin (5 x + y)$ .

$y' = \frac{5 \sin (5 x + y)}{- (\sin (5 x + y)) - 1}$

Step 5.6.3.1.2

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

$y' = \frac{5 \sin (5 x + y)}{- (\sin (5 x + y)) - 1 (1)}$

Step 5.6.3.1.3

Factor $- 1$ out of $- (\sin (5 x + y)) - 1 (1)$ .

$y' = \frac{5 \sin (5 x + y)}{- (\sin (5 x + y) + 1)}$

Step 5.6.3.2

Move the negative in front of the fraction.

$y' = - \frac{5 \sin (5 x + y)}{\sin (5 x + y) + 1}$

Step 6

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

$\frac{d y}{d x} = - \frac{5 \sin (5 x + y)}{\sin (5 x + y) + 1}$