Calculus Examples

$\tan (x y) = x - 1$ , $(1, 0)$

Step 1

Find the first derivative and evaluate at $x = 1$ and $y = 0$ to find the slope of the tangent line.

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Step 1.1

Differentiate both sides of the equation.

$\frac{d}{d x} (\tan (x y)) = \frac{d}{d x} (x - 1)$

Step 1.2

Differentiate the left side of the equation.

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Step 1.2.1

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

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Step 1.2.1.1

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

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

Step 1.2.1.2

The derivative of $\tan (u)$ with respect to $u$ is $\sec^{2} (u)$ .

$\sec^{2} (u) \frac{d}{d x} [x y]$

Step 1.2.1.3

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

$\sec^{2} (x y) \frac{d}{d x} [x y]$

Step 1.2.2

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 $g (x) = y$ .

$\sec^{2} (x y) (x \frac{d}{d x} [y] + y \frac{d}{d x} [x])$

Step 1.2.3

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

$\sec^{2} (x y) (x y' + y \frac{d}{d x} [x])$

Step 1.2.4

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

$\sec^{2} (x y) (x y' + y \cdot 1)$

Step 1.2.5

Multiply $y$ by $1$ .

$\sec^{2} (x y) (x y' + y)$

Step 1.2.6

Simplify.

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Step 1.2.6.1

Apply the distributive property.

$\sec^{2} (x y) (x y') + \sec^{2} (x y) y$

Step 1.2.6.2

Reorder terms.

$x \sec^{2} (x y) y' + y \sec^{2} (x y)$

Step 1.3

Differentiate the right side of the equation.

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Step 1.3.1

By the Sum Rule, the derivative of $x - 1$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [- 1]$ .

$\frac{d}{d x} [x] + \frac{d}{d x} [- 1]$

Step 1.3.2

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

$1 + \frac{d}{d x} [- 1]$

Step 1.3.3

Since $- 1$ is constant with respect to $x$ , the derivative of $- 1$ with respect to $x$ is $0$ .

$1 + 0$

Step 1.3.4

Add $1$ and $0$ .

$1$

Step 1.4

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

$x \sec^{2} (x y) y' + y \sec^{2} (x y) = 1$

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 $x \sec^{2} (x y) y' + y \sec^{2} (x y)$ .

$x y' \sec^{2} (x y) + y \sec^{2} (x y) = 1$

Step 1.5.2

Subtract $y \sec^{2} (x y)$ from both sides of the equation.

$x y' \sec^{2} (x y) = 1 - y \sec^{2} (x y)$

Step 1.5.3

Divide each term in $x y' \sec^{2} (x y) = 1 - y \sec^{2} (x y)$ by $x \sec^{2} (x y)$ and simplify.

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Step 1.5.3.1

Divide each term in $x y' \sec^{2} (x y) = 1 - y \sec^{2} (x y)$ by $x \sec^{2} (x y)$ .

$\frac{x y' \sec^{2} (x y)}{x \sec^{2} (x y)} = \frac{1}{x \sec^{2} (x y)} + \frac{- y \sec^{2} (x y)}{x \sec^{2} (x y)}$

Step 1.5.3.2

Simplify the left side.

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Step 1.5.3.2.1

Cancel the common factor of $x$ .

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Step 1.5.3.2.1.1

Cancel the common factor.

$\frac{x y' \sec^{2} (x y)}{x \sec^{2} (x y)} = \frac{1}{x \sec^{2} (x y)} + \frac{- y \sec^{2} (x y)}{x \sec^{2} (x y)}$

Step 1.5.3.2.1.2

Rewrite the expression.

$\frac{y' \sec^{2} (x y)}{\sec^{2} (x y)} = \frac{1}{x \sec^{2} (x y)} + \frac{- y \sec^{2} (x y)}{x \sec^{2} (x y)}$

Step 1.5.3.2.2

Cancel the common factor of $\sec^{2} (x y)$ .

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Step 1.5.3.2.2.1

Cancel the common factor.

$\frac{y' \sec^{2} (x y)}{\sec^{2} (x y)} = \frac{1}{x \sec^{2} (x y)} + \frac{- y \sec^{2} (x y)}{x \sec^{2} (x y)}$

Step 1.5.3.2.2.2

Divide $y'$ by $1$ .

$y' = \frac{1}{x \sec^{2} (x y)} + \frac{- y \sec^{2} (x y)}{x \sec^{2} (x y)}$

Step 1.5.3.3

Simplify the right side.

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Step 1.5.3.3.1

Simplify each term.

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Step 1.5.3.3.1.1

Cancel the common factor of $\sec^{2} (x y)$ .

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Step 1.5.3.3.1.1.1

Cancel the common factor.

$y' = \frac{1}{x \sec^{2} (x y)} + \frac{- y \sec^{2} (x y)}{x \sec^{2} (x y)}$

Step 1.5.3.3.1.1.2

Rewrite the expression.

$y' = \frac{1}{x \sec^{2} (x y)} + \frac{- y}{x}$

Step 1.5.3.3.1.2

Move the negative in front of the fraction.

$y' = \frac{1}{x \sec^{2} (x y)} - \frac{y}{x}$

Step 1.6

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

$\frac{d y}{d x} = \frac{1}{x \sec^{2} (x y)} - \frac{y}{x}$

Step 1.7

Evaluate at $x = 1$ and $y = 0$ .

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Step 1.7.1

Replace the variable $x$ with $1$ in the expression.

$\frac{1}{(1) \sec^{2} ((1) \cdot y)} - \frac{y}{1}$

Step 1.7.2

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

$\frac{1}{(1) \sec^{2} ((1) \cdot (0))} - \frac{0}{1}$

Step 1.7.3

Simplify each term.

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Step 1.7.3.1

Cancel the common factor of $1$ .

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Step 1.7.3.1.1

Cancel the common factor.

$\frac{1}{1 \sec^{2} ((1) \cdot (0))} - \frac{0}{1}$

Step 1.7.3.1.2

Rewrite the expression.

$\frac{1}{\sec^{2} ((1) \cdot (0))} - \frac{0}{1}$

Step 1.7.3.2

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

$\frac{1^{2}}{\sec^{2} ((1) \cdot (0))} - \frac{0}{1}$

Step 1.7.3.3

Rewrite $\frac{1^{2}}{\sec^{2} ((1) \cdot (0))}$ as ${(\frac{1}{\sec ((1) \cdot (0))})}^{2}$ .

${(\frac{1}{\sec ((1) \cdot (0))})}^{2} - \frac{0}{1}$

Step 1.7.3.4

Rewrite $\sec ((1) \cdot (0))$ in terms of sines and cosines.

${(\frac{1}{\frac{1}{\cos ((1) \cdot (0))}})}^{2} - \frac{0}{1}$

Step 1.7.3.5

Multiply by the reciprocal of the fraction to divide by $\frac{1}{\cos ((1) \cdot (0))}$ .

${(1 \cos ((1) \cdot (0)))}^{2} - \frac{0}{1}$

Step 1.7.3.6

Simplify.

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Step 1.7.3.6.1

Multiply $0$ by $1$ .

${(1 \cos (0))}^{2} - \frac{0}{1}$

Step 1.7.3.6.2

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

$\cos^{2} (0) - \frac{0}{1}$

Step 1.7.3.7

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

$1^{2} - \frac{0}{1}$

Step 1.7.3.8

One to any power is one.

$1 - \frac{0}{1}$

Step 1.7.3.9

Divide $0$ by $1$ .

$1 - 0$

Step 1.7.3.10

Multiply $- 1$ by $0$ .

$1 + 0$

Step 1.7.4

Add $1$ and $0$ .

$1$

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 $(1, 0)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

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

Step 2.2

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

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

Step 2.3

Solve for $y$ .

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Step 2.3.1

Add $y$ and $0$ .

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

Step 2.3.2

Multiply $x - 1$ by $1$ .

$y = x - 1$

Step 3