Calculus Examples

$y = \sec (x) + \tan (x)$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d y} (y) = \frac{d}{d y} (\sec (x) + \tan (x))$

Step 2

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

$1$

Step 3

Differentiate the right side of the equation.

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Step 3.1

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

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

Step 3.2

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

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Step 3.2.1

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

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Step 3.2.1.1

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

$\frac{d}{d u_{1}} [\sec (u_{1})] \frac{d}{d y} [x] + \frac{d}{d y} [\tan (x)]$

Step 3.2.1.2

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

$\sec (u_{1}) \tan (u_{1}) \frac{d}{d y} [x] + \frac{d}{d y} [\tan (x)]$

Step 3.2.1.3

Replace all occurrences of $u_{1}$ with $x$ .

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

Step 3.2.2

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

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

Step 3.3

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

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Step 3.3.1

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

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Step 3.3.1.1

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

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

Step 3.3.1.2

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

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

Step 3.3.1.3

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

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

Step 3.3.2

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

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

Step 4

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

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

Step 5

Solve for $x'$ .

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Step 5.1

Rewrite the equation as $\sec (x) \tan (x) x' + \sec^{2} (x) x' = 1$ .

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

Step 5.2

Simplify the left side.

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Step 5.2.1

Reorder factors in $\sec (x) \tan (x) x' + \sec^{2} (x) x'$ .

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

Step 5.3

Factor $x' \sec (x)$ out of $x' \sec (x) \tan (x) + x' \sec^{2} (x)$ .

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Step 5.3.1

Factor $x' \sec (x)$ out of $x' \sec^{2} (x)$ .

$x' \sec (x) \tan (x) + x' \sec (x) \sec (x) = 1$

Step 5.3.2

Factor $x' \sec (x)$ out of $x' \sec (x) \tan (x) + x' \sec (x) \sec (x)$ .

$x' \sec (x) (\tan (x) + \sec (x)) = 1$

Step 5.4

Divide each term in $x' \sec (x) (\tan (x) + \sec (x)) = 1$ by $\sec (x) (\tan (x) + \sec (x))$ and simplify.

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Step 5.4.1

Divide each term in $x' \sec (x) (\tan (x) + \sec (x)) = 1$ by $\sec (x) (\tan (x) + \sec (x))$ .

$\frac{x' \sec (x) (\tan (x) + \sec (x))}{\sec (x) (\tan (x) + \sec (x))} = \frac{1}{\sec (x) (\tan (x) + \sec (x))}$

Step 5.4.2

Simplify the left side.

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Step 5.4.2.1

Cancel the common factor of $\sec (x)$ .

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Step 5.4.2.1.1

Cancel the common factor.

$\frac{x' \sec (x) (\tan (x) + \sec (x))}{\sec (x) (\tan (x) + \sec (x))} = \frac{1}{\sec (x) (\tan (x) + \sec (x))}$

Step 5.4.2.1.2

Rewrite the expression.

$\frac{x' (\tan (x) + \sec (x))}{\tan (x) + \sec (x)} = \frac{1}{\sec (x) (\tan (x) + \sec (x))}$

Step 5.4.2.2

Cancel the common factor of $\tan (x) + \sec (x)$ .

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Step 5.4.2.2.1

Cancel the common factor.

$\frac{x' (\tan (x) + \sec (x))}{\tan (x) + \sec (x)} = \frac{1}{\sec (x) (\tan (x) + \sec (x))}$

Step 5.4.2.2.2

Divide $x'$ by $1$ .

$x' = \frac{1}{\sec (x) (\tan (x) + \sec (x))}$

Step 5.4.3

Simplify the right side.

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Step 5.4.3.1

Separate fractions.

$x' = \frac{1}{\tan (x) + \sec (x)} \cdot \frac{1}{\sec (x)}$

Step 5.4.3.2

Rewrite $\sec (x)$ in terms of sines and cosines.

$x' = \frac{1}{\tan (x) + \sec (x)} \cdot \frac{1}{\frac{1}{\cos (x)}}$

Step 5.4.3.3

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

$x' = \frac{1}{\tan (x) + \sec (x)} (1 \cos (x))$

Step 5.4.3.4

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

$x' = \frac{1}{\tan (x) + \sec (x)} \cos (x)$

Step 5.4.3.5

Combine $\frac{1}{\tan (x) + \sec (x)}$ and $\cos (x)$ .

$x' = \frac{\cos (x)}{\tan (x) + \sec (x)}$

Step 6

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

$\frac{d x}{d y} = \frac{\cos (x)}{\tan (x) + \sec (x)}$