Calculus Examples

$8 y + \sin (y) = x$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (8 y + \sin (y)) = \frac{d}{d x} (x)$

Step 2

Differentiate the left side of the equation.

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Step 2.1

By the Sum Rule, the derivative of $8 y + \sin (y)$ with respect to $x$ is $\frac{d}{d x} [8 y] + \frac{d}{d x} [\sin (y)]$ .

$\frac{d}{d x} [8 y] + \frac{d}{d x} [\sin (y)]$

Step 2.2

Evaluate $\frac{d}{d x} [8 y]$ .

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Step 2.2.1

Since $8$ is constant with respect to $x$ , the derivative of $8 y$ with respect to $x$ is $8 \frac{d}{d x} [y]$ .

$8 \frac{d}{d x} [y] + \frac{d}{d x} [\sin (y)]$

Step 2.2.2

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

$8 y' + \frac{d}{d x} [\sin (y)]$

Step 2.3

Evaluate $\frac{d}{d x} [\sin (y)]$ .

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Step 2.3.1

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

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Step 2.3.1.1

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

$8 y' + \frac{d}{d u} [\sin (u)] \frac{d}{d x} [y]$

Step 2.3.1.2

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

$8 y' + \cos (u) \frac{d}{d x} [y]$

Step 2.3.1.3

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

$8 y' + \cos (y) \frac{d}{d x} [y]$

Step 2.3.2

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

$8 y' + \cos (y) y'$

Step 2.4

Reorder terms.

$\cos (y) y' + 8 y'$

Step 3

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

$1$

Step 4

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

$\cos (y) y' + 8 y' = 1$

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 $\cos (y) y' + 8 y'$ .

$y' \cos (y) + 8 y' = 1$

Step 5.2

Factor $y'$ out of $y' \cos (y) + 8 y'$ .

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Step 5.2.1

Factor $y'$ out of $y' \cos (y)$ .

$y' (\cos (y)) + 8 y' = 1$

Step 5.2.2

Factor $y'$ out of $8 y'$ .

$y' (\cos (y)) + y' \cdot 8 = 1$

Step 5.2.3

Factor $y'$ out of $y' (\cos (y)) + y' \cdot 8$ .

$y' (\cos (y) + 8) = 1$

Step 5.3

Divide each term in $y' (\cos (y) + 8) = 1$ by $\cos (y) + 8$ and simplify.

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Step 5.3.1

Divide each term in $y' (\cos (y) + 8) = 1$ by $\cos (y) + 8$ .

$\frac{y' (\cos (y) + 8)}{\cos (y) + 8} = \frac{1}{\cos (y) + 8}$

Step 5.3.2

Simplify the left side.

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Step 5.3.2.1

Cancel the common factor of $\cos (y) + 8$ .

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Step 5.3.2.1.1

Cancel the common factor.

$\frac{y' (\cos (y) + 8)}{\cos (y) + 8} = \frac{1}{\cos (y) + 8}$

Step 5.3.2.1.2

Divide $y'$ by $1$ .

$y' = \frac{1}{\cos (y) + 8}$

Step 6

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

$\frac{d y}{d x} = \frac{1}{\cos (y) + 8}$