Calculus Examples

$y e^{\sin (x)} = x \cos (y)$ , $(0, 0)$

Step 1

Find the first derivative and evaluate at $x = 0$ and $y = 0$ to find the slope of the tangent line.

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Step 1.1

Differentiate both sides of the equation.

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

Step 1.2

Differentiate the left side of the equation.

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Step 1.2.1

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

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

Step 1.2.2

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

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Step 1.2.2.1

To apply the Chain Rule, set $u$ as $\sin (x)$ .

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

Step 1.2.2.2

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

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

Step 1.2.2.3

Replace all occurrences of $u$ with $\sin (x)$ .

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

Step 1.2.3

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

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

Step 1.2.4

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

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

Step 1.3

Differentiate the right side of the equation.

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Step 1.3.1

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

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

Step 1.3.2

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 $g (x) = y$ .

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Step 1.3.2.1

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

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

Step 1.3.2.2

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

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

Step 1.3.2.3

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

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

Step 1.3.3

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

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

Step 1.3.4

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

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

Step 1.3.5

Simplify the expression.

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Step 1.3.5.1

Multiply $\cos (y)$ by $1$ .

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

Step 1.3.5.2

Reorder terms.

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

Step 1.4

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

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

Step 1.5

Solve for $y'$ .

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Step 1.5.1

Simplify the left side.

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Step 1.5.1.1

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

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

Step 1.5.2

Simplify the right side.

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Step 1.5.2.1

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

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

Step 1.5.3

Add $x y' \sin (y)$ to both sides of the equation.

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

Step 1.5.4

Subtract $y e^{\sin (x)} \cos (x)$ from both sides of the equation.

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

Step 1.5.5

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

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Step 1.5.5.1

Factor $y'$ out of $x y' \sin (y)$ .

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

Step 1.5.5.2

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

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

Step 1.5.6

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

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Step 1.5.6.1

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

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

Step 1.5.6.2

Simplify the left side.

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Step 1.5.6.2.1

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

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Step 1.5.6.2.1.1

Cancel the common factor.

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

Step 1.5.6.2.1.2

Divide $y'$ by $1$ .

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

Step 1.5.6.3

Simplify the right side.

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Step 1.5.6.3.1

Move the negative in front of the fraction.

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

Step 1.5.6.3.2

Combine the numerators over the common denominator.

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

Step 1.6

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

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

Step 1.7

Evaluate at $x = 0$ and $y = 0$ .

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Step 1.7.1

Replace the variable $x$ with $0$ in the expression.

$\frac{\cos (y) - y e^{\sin (0)} \cos (0)}{e^{\sin (0)} + (0) \sin (y)}$

Step 1.7.2

Replace the variable $y$ with $0$ in the expression.

$\frac{\cos (0) + 0 \cdot (e^{\sin (0)} \cos (0))}{e^{\sin (0)} + (0) \sin (0)}$

Step 1.7.3

Simplify the numerator.

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Step 1.7.3.1

The exact value of $\cos (0)$ is $1$ .

$\frac{1 + 0 \cdot (e^{\sin (0)} \cos (0))}{e^{\sin (0)} + (0) \sin (0)}$

Step 1.7.3.2

The exact value of $\sin (0)$ is $0$ .

$\frac{1 + 0 \cdot (e^{0} \cos (0))}{e^{\sin (0)} + (0) \sin (0)}$

Step 1.7.3.3

Anything raised to $0$ is $1$ .

$\frac{1 + 0 \cdot (1 \cos (0))}{e^{\sin (0)} + (0) \sin (0)}$

Step 1.7.3.4

Multiply $\cos (0)$ by $1$ .

$\frac{1 + 0 \cdot \cos (0)}{e^{\sin (0)} + (0) \sin (0)}$

Step 1.7.3.5

The exact value of $\cos (0)$ is $1$ .

$\frac{1 + 0 \cdot 1}{e^{\sin (0)} + (0) \sin (0)}$

Step 1.7.3.6

Multiply $0$ by $1$ .

$\frac{1 + 0}{e^{\sin (0)} + (0) \sin (0)}$

Step 1.7.3.7

Add $1$ and $0$ .

$\frac{1}{e^{\sin (0)} + (0) \sin (0)}$

Step 1.7.4

Simplify the denominator.

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Step 1.7.4.1

The exact value of $\sin (0)$ is $0$ .

$\frac{1}{e^{0} + (0) \sin (0)}$

Step 1.7.4.2

Anything raised to $0$ is $1$ .

$\frac{1}{1 + (0) \sin (0)}$

Step 1.7.4.3

The exact value of $\sin (0)$ is $0$ .

$\frac{1}{1 + 0 \cdot 0}$

Step 1.7.4.4

Multiply $0$ by $0$ .

$\frac{1}{1 + 0}$

Step 1.7.4.5

Add $1$ and $0$ .

$\frac{1}{1}$

Step 1.7.5

Cancel the common factor of $1$ .

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Step 1.7.5.1

Cancel the common factor.

$\frac{1}{1}$

Step 1.7.5.2

Rewrite the expression.

$1$

Step 2

Plug the slope and point values into the point-slope formula and solve for $y$ .

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Step 2.1

Use the slope $1$ and a given point $(0, 0)$ to substitute for $x_{1}$ and $y_{1}$ in the point-slope form $y - y_{1} = m (x - x_{1})$ , which is derived from the slope equation $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (0) = 1 \cdot (x - (0))$

Step 2.2

Simplify the equation and keep it in point-slope form.

$y + 0 = 1 \cdot (x + 0)$

Step 2.3

Solve for $y$ .

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Step 2.3.1

Add $y$ and $0$ .

$y = 1 \cdot (x + 0)$

Step 2.3.2

Simplify $1 \cdot (x + 0)$ .

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Step 2.3.2.1

Multiply $x + 0$ by $1$ .

$y = x + 0$

Step 2.3.2.2

Add $x$ and $0$ .

$y = x$

Step 3