Calculus Examples

$x - \cos (7 y) = 0$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (x - \cos (7 y)) = \frac{d}{d x} (0)$

Step 2

Differentiate the left side of the equation.

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Step 2.1

Differentiate.

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Step 2.1.1

By the Sum Rule, the derivative of $x - \cos (7 y)$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [- \cos (7 y)]$ .

$\frac{d}{d x} [x] + \frac{d}{d x} [- \cos (7 y)]$

Step 2.1.2

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

$1 + \frac{d}{d x} [- \cos (7 y)]$

Step 2.2

Evaluate $\frac{d}{d x} [- \cos (7 y)]$ .

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Step 2.2.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- \cos (7 y)$ with respect to $x$ is $- \frac{d}{d x} [\cos (7 y)]$ .

$1 - \frac{d}{d x} [\cos (7 y)]$

Step 2.2.2

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 $g (x) = 7 y$ .

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Step 2.2.2.1

To apply the Chain Rule, set $u$ as $7 y$ .

$1 - (\frac{d}{d u} [\cos (u)] \frac{d}{d x} [7 y])$

Step 2.2.2.2

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

$1 - (- \sin (u) \frac{d}{d x} [7 y])$

Step 2.2.2.3

Replace all occurrences of $u$ with $7 y$ .

$1 - (- \sin (7 y) \frac{d}{d x} [7 y])$

Step 2.2.3

Since $7$ is constant with respect to $x$ , the derivative of $7 y$ with respect to $x$ is $7 \frac{d}{d x} [y]$ .

$1 - (- \sin (7 y) (7 \frac{d}{d x} [y]))$

Step 2.2.4

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

$1 - (- \sin (7 y) (7 y'))$

Step 2.2.5

Multiply $7$ by $- 1$ .

$1 - (- 7 \sin (7 y) y')$

Step 2.2.6

Multiply $- 7$ by $- 1$ .

$1 + 7 \sin (7 y) y'$

Step 2.3

Reorder terms.

$7 \sin (7 y) y' + 1$

Step 3

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

$0$

Step 4

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

$7 \sin (7 y) y' + 1 = 0$

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 $7 \sin (7 y) y' + 1$ .

$7 y' \sin (7 y) + 1 = 0$

Step 5.2

Subtract $1$ from both sides of the equation.

$7 y' \sin (7 y) = - 1$

Step 5.3

Divide each term in $7 y' \sin (7 y) = - 1$ by $7 \sin (7 y)$ and simplify.

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Step 5.3.1

Divide each term in $7 y' \sin (7 y) = - 1$ by $7 \sin (7 y)$ .

$\frac{7 y' \sin (7 y)}{7 \sin (7 y)} = \frac{- 1}{7 \sin (7 y)}$

Step 5.3.2

Simplify the left side.

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Step 5.3.2.1

Cancel the common factor of $7$ .

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Step 5.3.2.1.1

Cancel the common factor.

$\frac{7 y' \sin (7 y)}{7 \sin (7 y)} = \frac{- 1}{7 \sin (7 y)}$

Step 5.3.2.1.2

Rewrite the expression.

$\frac{y' \sin (7 y)}{\sin (7 y)} = \frac{- 1}{7 \sin (7 y)}$

Step 5.3.2.2

Cancel the common factor of $\sin (7 y)$ .

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Step 5.3.2.2.1

Cancel the common factor.

$\frac{y' \sin (7 y)}{\sin (7 y)} = \frac{- 1}{7 \sin (7 y)}$

Step 5.3.2.2.2

Divide $y'$ by $1$ .

$y' = \frac{- 1}{7 \sin (7 y)}$

Step 5.3.3

Simplify the right side.

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Step 5.3.3.1

Separate fractions.

$y' = \frac{- 1}{7} \cdot \frac{1}{\sin (7 y)}$

Step 5.3.3.2

Convert from $\frac{1}{\sin (7 y)}$ to $\csc (7 y)$ .

$y' = \frac{- 1}{7} \csc (7 y)$

Step 5.3.3.3

Move the negative in front of the fraction.

$y' = - \frac{1}{7} \csc (7 y)$

Step 5.3.3.4

Combine $\csc (7 y)$ and $\frac{1}{7}$ .

$y' = - \frac{\csc (7 y)}{7}$

Step 6

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

$\frac{d y}{d x} = - \frac{\csc (7 y)}{7}$