Calculus Examples

$7 \cos (7 x) \sin (5 y) = 3$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (7 \cos (7 x) \sin (5 y)) = \frac{d}{d x} (3)$

Step 2

Differentiate the left side of the equation.

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Step 2.1

Since $7$ is constant with respect to $x$ , the derivative of $7 \cos (7 x) \sin (5 y)$ with respect to $x$ is $7 \frac{d}{d x} [\cos (7 x) \sin (5 y)]$ .

$7 \frac{d}{d x} [\cos (7 x) \sin (5 y)]$

Step 2.2

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = \cos (7 x)$ and $g (x) = \sin (5 y)$ .

$7 (\cos (7 x) \frac{d}{d x} [\sin (5 y)] + \sin (5 y) \frac{d}{d x} [\cos (7 x)])$

Step 2.3

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 $g (x) = 5 y$ .

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Step 2.3.1

To apply the Chain Rule, set $u_{1}$ as $5 y$ .

$7 (\cos (7 x) (\frac{d}{d u_{1}} [\sin (u_{1})] \frac{d}{d x} [5 y]) + \sin (5 y) \frac{d}{d x} [\cos (7 x)])$

Step 2.3.2

The derivative of $\sin (u_{1})$ with respect to $u_{1}$ is $\cos (u_{1})$ .

$7 (\cos (7 x) (\cos (u_{1}) \frac{d}{d x} [5 y]) + \sin (5 y) \frac{d}{d x} [\cos (7 x)])$

Step 2.3.3

Replace all occurrences of $u_{1}$ with $5 y$ .

$7 (\cos (7 x) (\cos (5 y) \frac{d}{d x} [5 y]) + \sin (5 y) \frac{d}{d x} [\cos (7 x)])$

Step 2.4

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

$7 (\cos (7 x) \cos (5 y) (5 \frac{d}{d x} [y]) + \sin (5 y) \frac{d}{d x} [\cos (7 x)])$

Step 2.5

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

$7 (\cos (7 x) \cos (5 y) (5 y') + \sin (5 y) \frac{d}{d x} [\cos (7 x)])$

Step 2.6

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 $g (x) = 7 x$ .

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Step 2.6.1

To apply the Chain Rule, set $u_{2}$ as $7 x$ .

$7 (\cos (7 x) \cos (5 y) (5 y') + \sin (5 y) (\frac{d}{d u_{2}} [\cos (u_{2})] \frac{d}{d x} [7 x]))$

Step 2.6.2

The derivative of $\cos (u_{2})$ with respect to $u_{2}$ is $- \sin (u_{2})$ .

$7 (\cos (7 x) \cos (5 y) (5 y') + \sin (5 y) (- \sin (u_{2}) \frac{d}{d x} [7 x]))$

Step 2.6.3

Replace all occurrences of $u_{2}$ with $7 x$ .

$7 (\cos (7 x) \cos (5 y) (5 y') + \sin (5 y) (- \sin (7 x) \frac{d}{d x} [7 x]))$

Step 2.7

Differentiate.

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Step 2.7.1

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

$7 (\cos (7 x) \cos (5 y) (5 y') + \sin (5 y) (- \sin (7 x) (7 \frac{d}{d x} [x])))$

Step 2.7.2

Multiply $7$ by $- 1$ .

$7 (\cos (7 x) \cos (5 y) (5 y') + \sin (5 y) (- 7 \sin (7 x) \frac{d}{d x} [x]))$

Step 2.7.3

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

$7 (\cos (7 x) \cos (5 y) (5 y') + \sin (5 y) (- 7 \sin (7 x) \cdot 1))$

Step 2.7.4

Multiply $- 7$ by $1$ .

$7 (\cos (7 x) \cos (5 y) (5 y') + \sin (5 y) (- 7 \sin (7 x)))$

Step 2.8

Simplify.

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Step 2.8.1

Apply the distributive property.

$7 (\cos (7 x) \cos (5 y) (5 y')) + 7 (\sin (5 y) (- 7 \sin (7 x)))$

Step 2.8.2

Combine terms.

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Step 2.8.2.1

Multiply $5$ by $7$ .

$35 (\cos (7 x) \cos (5 y) (y')) + 7 (\sin (5 y) (- 7 \sin (7 x)))$

Step 2.8.2.2

Multiply $- 7$ by $7$ .

$35 \cos (7 x) \cos (5 y) y' - 49 \sin (5 y) \sin (7 x)$

Step 2.8.3

Reorder terms.

$35 \cos (5 y) \cos (7 x) y' - 49 \sin (5 y) \sin (7 x)$

Step 3

Since $3$ is constant with respect to $x$ , the derivative of $3$ with respect to $x$ is $0$ .

$0$

Step 4

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

$35 \cos (5 y) \cos (7 x) y' - 49 \sin (5 y) \sin (7 x) = 0$

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 $35 \cos (5 y) \cos (7 x) y' - 49 \sin (5 y) \sin (7 x)$ .

$35 y' \cos (5 y) \cos (7 x) - 49 \sin (5 y) \sin (7 x) = 0$

Step 5.2

Add $49 \sin (5 y) \sin (7 x)$ to both sides of the equation.

$35 y' \cos (5 y) \cos (7 x) = 49 \sin (5 y) \sin (7 x)$

Step 5.3

Divide each term in $35 y' \cos (5 y) \cos (7 x) = 49 \sin (5 y) \sin (7 x)$ by $35 \cos (5 y) \cos (7 x)$ and simplify.

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Step 5.3.1

Divide each term in $35 y' \cos (5 y) \cos (7 x) = 49 \sin (5 y) \sin (7 x)$ by $35 \cos (5 y) \cos (7 x)$ .

$\frac{35 y' \cos (5 y) \cos (7 x)}{35 \cos (5 y) \cos (7 x)} = \frac{49 \sin (5 y) \sin (7 x)}{35 \cos (5 y) \cos (7 x)}$

Step 5.3.2

Simplify the left side.

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Step 5.3.2.1

Cancel the common factor of $35$ .

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Step 5.3.2.1.1

Cancel the common factor.

$\frac{35 y' \cos (5 y) \cos (7 x)}{35 \cos (5 y) \cos (7 x)} = \frac{49 \sin (5 y) \sin (7 x)}{35 \cos (5 y) \cos (7 x)}$

Step 5.3.2.1.2

Rewrite the expression.

$\frac{y' \cos (5 y) \cos (7 x)}{\cos (5 y) \cos (7 x)} = \frac{49 \sin (5 y) \sin (7 x)}{35 \cos (5 y) \cos (7 x)}$

Step 5.3.2.2

Cancel the common factor of $\cos (5 y)$ .

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Step 5.3.2.2.1

Cancel the common factor.

$\frac{y' \cos (5 y) \cos (7 x)}{\cos (5 y) \cos (7 x)} = \frac{49 \sin (5 y) \sin (7 x)}{35 \cos (5 y) \cos (7 x)}$

Step 5.3.2.2.2

Rewrite the expression.

$\frac{y' \cos (7 x)}{\cos (7 x)} = \frac{49 \sin (5 y) \sin (7 x)}{35 \cos (5 y) \cos (7 x)}$

Step 5.3.2.3

Cancel the common factor of $\cos (7 x)$ .

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Step 5.3.2.3.1

Cancel the common factor.

$\frac{y' \cos (7 x)}{\cos (7 x)} = \frac{49 \sin (5 y) \sin (7 x)}{35 \cos (5 y) \cos (7 x)}$

Step 5.3.2.3.2

Divide $y'$ by $1$ .

$y' = \frac{49 \sin (5 y) \sin (7 x)}{35 \cos (5 y) \cos (7 x)}$

Step 5.3.3

Simplify the right side.

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Step 5.3.3.1

Cancel the common factor of $49$ and $35$ .

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Step 5.3.3.1.1

Factor $7$ out of $49 \sin (5 y) \sin (7 x)$ .

$y' = \frac{7 (7 \sin (5 y) \sin (7 x))}{35 \cos (5 y) \cos (7 x)}$

Step 5.3.3.1.2

Cancel the common factors.

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Step 5.3.3.1.2.1

Factor $7$ out of $35 \cos (5 y) \cos (7 x)$ .

$y' = \frac{7 (7 \sin (5 y) \sin (7 x))}{7 (5 \cos (5 y) \cos (7 x))}$

Step 5.3.3.1.2.2

Cancel the common factor.

$y' = \frac{7 (7 \sin (5 y) \sin (7 x))}{7 (5 \cos (5 y) \cos (7 x))}$

Step 5.3.3.1.2.3

Rewrite the expression.

$y' = \frac{7 \sin (5 y) \sin (7 x)}{5 \cos (5 y) \cos (7 x)}$

Step 5.3.3.2

Separate fractions.

$y' = \frac{7 \sin (7 x)}{5 \cos (7 x)} \cdot \frac{\sin (5 y)}{\cos (5 y)}$

Step 5.3.3.3

Convert from $\frac{\sin (5 y)}{\cos (5 y)}$ to $\tan (5 y)$ .

$y' = \frac{7 \sin (7 x)}{5 \cos (7 x)} \tan (5 y)$

Step 5.3.3.4

Separate fractions.

$y' = \frac{7}{5} \cdot \frac{\sin (7 x)}{\cos (7 x)} \tan (5 y)$

Step 5.3.3.5

Convert from $\frac{\sin (7 x)}{\cos (7 x)}$ to $\tan (7 x)$ .

$y' = \frac{7}{5} \tan (7 x) \tan (5 y)$

Step 5.3.3.6

Combine $\frac{7}{5}$ and $\tan (7 x)$ .

$y' = \frac{7 \tan (7 x)}{5} \tan (5 y)$

Step 5.3.3.7

Combine $\frac{7 \tan (7 x)}{5}$ and $\tan (5 y)$ .

$y' = \frac{7 \tan (7 x) \tan (5 y)}{5}$

Step 6

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

$\frac{d y}{d x} = \frac{7 \tan (7 x) \tan (5 y)}{5}$