Calculus Examples

$e^{\cos (y)} = x^{9} \arctan (y)$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d c} (e^{\cos (y)}) = \frac{d}{d c} (x^{9} \arctan (y))$

Step 2

Differentiate the left side of the equation.

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Step 2.1

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

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Step 2.1.1

To apply the Chain Rule, set $u_{1}$ as $\cos (y)$ .

$\frac{d}{d u_{1}} [e^{u_{1}}] \frac{d}{d c} [\cos (y)]$

Step 2.1.2

Differentiate using the Exponential Rule which states that $\frac{d}{d u_{1}} [a^{u_{1}}]$ is $a^{u_{1}} \ln (a)$ where $a$ = $e$ .

$e^{u_{1}} \frac{d}{d c} [\cos (y)]$

Step 2.1.3

Replace all occurrences of $u_{1}$ with $\cos (y)$ .

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

Step 2.2

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

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Step 2.2.1

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

$e^{\cos (y)} (\frac{d}{d u_{2}} [\cos (u_{2})] \frac{d}{d c} [y])$

Step 2.2.2

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

$e^{\cos (y)} (- \sin (u_{2}) \frac{d}{d c} [y])$

Step 2.2.3

Replace all occurrences of $u_{2}$ with $y$ .

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

Step 2.3

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

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

Step 2.4

Reorder the factors of $e^{\cos (y)} (- \sin (y) y')$ .

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

Step 3

Differentiate the right side of the equation.

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Step 3.1

Since $x^{9}$ is constant with respect to $c$ , the derivative of $x^{9} \arctan (y)$ with respect to $c$ is $x^{9} \frac{d}{d c} [\arctan (y)]$ .

$x^{9} \frac{d}{d c} [\arctan (y)]$

Step 3.2

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

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Step 3.2.1

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

$x^{9} (\frac{d}{d u} [\arctan (u)] \frac{d}{d c} [y])$

Step 3.2.2

The derivative of $\arctan (u)$ with respect to $u$ is $\frac{1}{1 + u^{2}}$ .

$x^{9} (\frac{1}{1 + u^{2}} \frac{d}{d c} [y])$

Step 3.2.3

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

$x^{9} (\frac{1}{1 + y^{2}} \frac{d}{d c} [y])$

Step 3.3

Combine $\frac{1}{1 + y^{2}}$ and $x^{9}$ .

$\frac{x^{9}}{1 + y^{2}} \frac{d}{d c} [y]$

Step 3.4

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

$\frac{x^{9}}{1 + y^{2}} y'$

Step 3.5

Combine $\frac{x^{9}}{1 + y^{2}}$ and $y'$ .

$\frac{x^{9} y'}{1 + y^{2}}$

Step 3.6

Reorder terms.

$\frac{x^{9} y'}{y^{2} + 1}$

Step 4

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

$- e^{\cos (y)} \sin (y) y' = \frac{x^{9} y'}{y^{2} + 1}$

Step 5

Solve for $y'$ .

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Step 5.1

Multiply both sides by $y^{2} + 1$ .

$- e^{\cos (y)} \sin (y) y' (y^{2} + 1) = \frac{x^{9} y'}{y^{2} + 1} (y^{2} + 1)$

Step 5.2

Simplify.

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Step 5.2.1

Simplify the left side.

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Step 5.2.1.1

Simplify $- e^{\cos (y)} \sin (y) y' (y^{2} + 1)$ .

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Step 5.2.1.1.1

Apply the distributive property.

$- e^{\cos (y)} \sin (y) y' y^{2} - e^{\cos (y)} \sin (y) y' \cdot 1 = \frac{x^{9} y'}{y^{2} + 1} (y^{2} + 1)$

Step 5.2.1.1.2

Simplify the expression.

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Step 5.2.1.1.2.1

Multiply $- 1$ by $1$ .

$- e^{\cos (y)} \sin (y) y' y^{2} - e^{\cos (y)} \sin (y) y' = \frac{x^{9} y'}{y^{2} + 1} (y^{2} + 1)$

Step 5.2.1.1.2.2

Reorder factors in $- e^{\cos (y)} \sin (y) y' y^{2} - e^{\cos (y)} \sin (y) y'$ .

$- y' y^{2} e^{\cos (y)} \sin (y) - y' e^{\cos (y)} \sin (y) = \frac{x^{9} y'}{y^{2} + 1} (y^{2} + 1)$

Step 5.2.2

Simplify the right side.

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Step 5.2.2.1

Cancel the common factor of $y^{2} + 1$ .

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Step 5.2.2.1.1

Cancel the common factor.

$- y' y^{2} e^{\cos (y)} \sin (y) - y' e^{\cos (y)} \sin (y) = \frac{x^{9} y'}{y^{2} + 1} (y^{2} + 1)$

Step 5.2.2.1.2

Rewrite the expression.

$- y' y^{2} e^{\cos (y)} \sin (y) - y' e^{\cos (y)} \sin (y) = x^{9} y'$

Step 5.3

Solve for $y'$ .

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Step 5.3.1

Subtract $x^{9} y'$ from both sides of the equation.

$- y' y^{2} e^{\cos (y)} \sin (y) - y' e^{\cos (y)} \sin (y) - x^{9} y' = 0$

Step 5.3.2

Factor $y'$ out of $- y' y^{2} e^{\cos (y)} \sin (y) - y' e^{\cos (y)} \sin (y) - x^{9} y'$ .

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Step 5.3.2.1

Factor $y'$ out of $- y' y^{2} e^{\cos (y)} \sin (y)$ .

$y' (- 1 y^{2} e^{\cos (y)} \sin (y)) - y' e^{\cos (y)} \sin (y) - x^{9} y' = 0$

Step 5.3.2.2

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

$y' (- 1 y^{2} e^{\cos (y)} \sin (y)) + y' (- 1 e^{\cos (y)} \sin (y)) - x^{9} y' = 0$

Step 5.3.2.3

Factor $y'$ out of $- x^{9} y'$ .

$y' (- 1 y^{2} e^{\cos (y)} \sin (y)) + y' (- 1 e^{\cos (y)} \sin (y)) + y' (- x^{9}) = 0$

Step 5.3.2.4

Factor $y'$ out of $y' (- 1 y^{2} e^{\cos (y)} \sin (y)) + y' (- 1 e^{\cos (y)} \sin (y))$ .

$y' (- 1 y^{2} e^{\cos (y)} \sin (y) - 1 e^{\cos (y)} \sin (y)) + y' (- x^{9}) = 0$

Step 5.3.2.5

Factor $y'$ out of $y' (- 1 y^{2} e^{\cos (y)} \sin (y) - 1 e^{\cos (y)} \sin (y)) + y' (- x^{9})$ .

$y' (- 1 y^{2} e^{\cos (y)} \sin (y) - 1 e^{\cos (y)} \sin (y) - x^{9}) = 0$

Step 5.3.3

Rewrite $- 1 y^{2}$ as $- y^{2}$ .

$y' (- y^{2} e^{\cos (y)} \sin (y) - 1 e^{\cos (y)} \sin (y) - x^{9}) = 0$

Step 5.3.4

Rewrite $- 1 e^{\cos (y)}$ as $- e^{\cos (y)}$ .

$y' (- y^{2} e^{\cos (y)} \sin (y) - e^{\cos (y)} \sin (y) - x^{9}) = 0$

Step 5.3.5

Divide each term in $y' (- y^{2} e^{\cos (y)} \sin (y) - e^{\cos (y)} \sin (y) - x^{9}) = 0$ by $- y^{2} e^{\cos (y)} \sin (y) - e^{\cos (y)} \sin (y) - x^{9}$ and simplify.

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Step 5.3.5.1

Divide each term in $y' (- y^{2} e^{\cos (y)} \sin (y) - e^{\cos (y)} \sin (y) - x^{9}) = 0$ by $- y^{2} e^{\cos (y)} \sin (y) - e^{\cos (y)} \sin (y) - x^{9}$ .

$\frac{y' (- y^{2} e^{\cos (y)} \sin (y) - e^{\cos (y)} \sin (y) - x^{9})}{- y^{2} e^{\cos (y)} \sin (y) - e^{\cos (y)} \sin (y) - x^{9}} = \frac{0}{- y^{2} e^{\cos (y)} \sin (y) - e^{\cos (y)} \sin (y) - x^{9}}$

Step 5.3.5.2

Simplify the left side.

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Step 5.3.5.2.1

Cancel the common factor of $- y^{2} e^{\cos (y)} \sin (y) - e^{\cos (y)} \sin (y) - x^{9}$ .

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Step 5.3.5.2.1.1

Cancel the common factor.

Step 5.3.5.2.1.2

Divide $y'$ by $1$ .

$y' = \frac{0}{- y^{2} e^{\cos (y)} \sin (y) - e^{\cos (y)} \sin (y) - x^{9}}$

Step 5.3.5.3

Simplify the right side.

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Step 5.3.5.3.1

Divide $0$ by $- y^{2} e^{\cos (y)} \sin (y) - e^{\cos (y)} \sin (y) - x^{9}$ .

$y' = 0$

Step 6

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

$\frac{d y}{d c} = 0$