Calculus Examples

$e^{y} \cos (x) = 6 + \sin (x y)$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (e^{y} \cos (x)) = \frac{d}{d x} (6 + \sin (x y))$

Step 2

Differentiate the left side of the equation.

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Step 2.1

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 $f (x) = e^{y}$ and $g (x) = \cos (x)$ .

$e^{y} \frac{d}{d x} [\cos (x)] + \cos (x) \frac{d}{d x} [e^{y}]$

Step 2.2

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

$e^{y} (- \sin (x)) + \cos (x) \frac{d}{d x} [e^{y}]$

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 $f (x) = e^{x}$ and $g (x) = y$ .

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Step 2.3.1

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

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

Step 2.3.2

Differentiate using the Exponential Rule which states that $\frac{d}{d u} [a^{u}]$ is $a^{u} \ln (a)$ where $a$ = $e$ .

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

Step 2.3.3

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

$e^{y} (- \sin (x)) + \cos (x) (e^{y} \frac{d}{d x} [y])$

Step 2.4

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

$e^{y} (- \sin (x)) + \cos (x) e^{y} y'$

Step 2.5

Reorder terms.

$- e^{y} \sin (x) + e^{y} \cos (x) y'$

Step 3

Differentiate the right side of the equation.

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Step 3.1

Differentiate.

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Step 3.1.1

By the Sum Rule, the derivative of $6 + \sin (x y)$ with respect to $x$ is $\frac{d}{d x} [6] + \frac{d}{d x} [\sin (x y)]$ .

$\frac{d}{d x} [6] + \frac{d}{d x} [\sin (x y)]$

Step 3.1.2

Since $6$ is constant with respect to $x$ , the derivative of $6$ with respect to $x$ is $0$ .

$0 + \frac{d}{d x} [\sin (x y)]$

Step 3.2

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

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Step 3.2.1

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

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Step 3.2.1.1

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

$0 + \frac{d}{d u} [\sin (u)] \frac{d}{d x} [x y]$

Step 3.2.1.2

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

$0 + \cos (u) \frac{d}{d x} [x y]$

Step 3.2.1.3

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

$0 + \cos (x y) \frac{d}{d x} [x y]$

Step 3.2.2

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 $f (x) = x$ and $g (x) = y$ .

$0 + \cos (x y) (x \frac{d}{d x} [y] + y \frac{d}{d x} [x])$

Step 3.2.3

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

$0 + \cos (x y) (x y' + y \frac{d}{d x} [x])$

Step 3.2.4

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

$0 + \cos (x y) (x y' + y \cdot 1)$

Step 3.2.5

Multiply $y$ by $1$ .

$0 + \cos (x y) (x y' + y)$

Step 3.3

Simplify.

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Step 3.3.1

Apply the distributive property.

$0 + \cos (x y) (x y') + \cos (x y) y$

Step 3.3.2

Add $0$ and $\cos (x y) (x y') + \cos (x y) y$ .

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

Step 3.3.3

Reorder terms.

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

Step 4

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

$- e^{y} \sin (x) + e^{y} \cos (x) y' = x \cos (x y) y' + y \cos (x y)$

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 $- e^{y} \sin (x) + e^{y} \cos (x) y'$ .

$- e^{y} \sin (x) + y' e^{y} \cos (x) = x \cos (x y) y' + y \cos (x y)$

Step 5.2

Simplify the right side.

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Step 5.2.1

Reorder factors in $x \cos (x y) y' + y \cos (x y)$ .

$- e^{y} \sin (x) + y' e^{y} \cos (x) = x y' \cos (x y) + y \cos (x y)$

Step 5.3

Subtract $x y' \cos (x y)$ from both sides of the equation.

$- e^{y} \sin (x) + y' e^{y} \cos (x) - x y' \cos (x y) = y \cos (x y)$

Step 5.4

Add $e^{y} \sin (x)$ to both sides of the equation.

$y' e^{y} \cos (x) - x y' \cos (x y) = y \cos (x y) + e^{y} \sin (x)$

Step 5.5

Factor $y'$ out of $y' e^{y} \cos (x) - x y' \cos (x y)$ .

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Step 5.5.1

Factor $y'$ out of $y' e^{y} \cos (x)$ .

$y' (e^{y} \cos (x)) - x y' \cos (x y) = y \cos (x y) + e^{y} \sin (x)$

Step 5.5.2

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

$y' (e^{y} \cos (x)) + y' (- x \cos (x y)) = y \cos (x y) + e^{y} \sin (x)$

Step 5.5.3

Factor $y'$ out of $y' (e^{y} \cos (x)) + y' (- x \cos (x y))$ .

$y' (e^{y} \cos (x) - x \cos (x y)) = y \cos (x y) + e^{y} \sin (x)$

Step 5.6

Divide each term in $y' (e^{y} \cos (x) - x \cos (x y)) = y \cos (x y) + e^{y} \sin (x)$ by $e^{y} \cos (x) - x \cos (x y)$ and simplify.

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Step 5.6.1

Divide each term in $y' (e^{y} \cos (x) - x \cos (x y)) = y \cos (x y) + e^{y} \sin (x)$ by $e^{y} \cos (x) - x \cos (x y)$ .

$\frac{y' (e^{y} \cos (x) - x \cos (x y))}{e^{y} \cos (x) - x \cos (x y)} = \frac{y \cos (x y)}{e^{y} \cos (x) - x \cos (x y)} + \frac{e^{y} \sin (x)}{e^{y} \cos (x) - x \cos (x y)}$

Step 5.6.2

Simplify the left side.

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Step 5.6.2.1

Cancel the common factor of $e^{y} \cos (x) - x \cos (x y)$ .

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Step 5.6.2.1.1

Cancel the common factor.

$\frac{y' (e^{y} \cos (x) - x \cos (x y))}{e^{y} \cos (x) - x \cos (x y)} = \frac{y \cos (x y)}{e^{y} \cos (x) - x \cos (x y)} + \frac{e^{y} \sin (x)}{e^{y} \cos (x) - x \cos (x y)}$

Step 5.6.2.1.2

Divide $y'$ by $1$ .

$y' = \frac{y \cos (x y)}{e^{y} \cos (x) - x \cos (x y)} + \frac{e^{y} \sin (x)}{e^{y} \cos (x) - x \cos (x y)}$

Step 5.6.3

Simplify the right side.

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Step 5.6.3.1

Combine the numerators over the common denominator.

$y' = \frac{y \cos (x y) + e^{y} \sin (x)}{e^{y} \cos (x) - x \cos (x y)}$

Step 6

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

$\frac{d y}{d x} = \frac{y \cos (x y) + e^{y} \sin (x)}{e^{y} \cos (x) - x \cos (x y)}$