Calculus Examples

$y = {(- 1 - \frac{\csc (x)}{2} - \frac{x^{2}}{4})}^{2}$

Step 1

Differentiate using the chain rule, which states that $\frac{d}{d ?} [f (g (?))]$ is $f' (g (?)) g' (?)$ where $f (?) = ?^{2}$ and $g (?) = - 1 - \frac{\csc (x)}{2} - \frac{x^{2}}{4}$ .

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Step 1.1

To apply the Chain Rule, set $u$ as $- 1 - \frac{\csc (x)}{2} - \frac{x^{2}}{4}$ .

$\frac{d}{d u} [u^{2}] \frac{d}{d ?} [- 1 - \frac{\csc (x)}{2} - \frac{x^{2}}{4}]$

Step 1.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = 2$ .

$2 u \frac{d}{d ?} [- 1 - \frac{\csc (x)}{2} - \frac{x^{2}}{4}]$

Step 1.3

Replace all occurrences of $u$ with $- 1 - \frac{\csc (x)}{2} - \frac{x^{2}}{4}$ .

$2 (- 1 - \frac{\csc (x)}{2} - \frac{x^{2}}{4}) \frac{d}{d ?} [- 1 - \frac{\csc (x)}{2} - \frac{x^{2}}{4}]$

Step 2

Differentiate.

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Step 2.1

By the Sum Rule, the derivative of $- 1 - \frac{\csc (x)}{2} - \frac{x^{2}}{4}$ with respect to $?$ is $\frac{d}{d ?} [- 1] + \frac{d}{d ?} [- \frac{\csc (x)}{2}] + \frac{d}{d ?} [- \frac{x^{2}}{4}]$ .

$2 (- 1 - \frac{\csc (x)}{2} - \frac{x^{2}}{4}) (\frac{d}{d ?} [- 1] + \frac{d}{d ?} [- \frac{\csc (x)}{2}] + \frac{d}{d ?} [- \frac{x^{2}}{4}])$

Step 2.2

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

$2 (- 1 - \frac{\csc (x)}{2} - \frac{x^{2}}{4}) (0 + \frac{d}{d ?} [- \frac{\csc (x)}{2}] + \frac{d}{d ?} [- \frac{x^{2}}{4}])$

Step 2.3

Since $- \frac{\csc (x)}{2}$ is constant with respect to $?$ , the derivative of $- \frac{\csc (x)}{2}$ with respect to $?$ is $0$ .

$2 (- 1 - \frac{\csc (x)}{2} - \frac{x^{2}}{4}) (0 + 0 + \frac{d}{d ?} [- \frac{x^{2}}{4}])$

Step 2.4

Since $- \frac{x^{2}}{4}$ is constant with respect to $?$ , the derivative of $- \frac{x^{2}}{4}$ with respect to $?$ is $0$ .

$2 (- 1 - \frac{\csc (x)}{2} - \frac{x^{2}}{4}) (0 + 0 + 0)$

Step 3

Combine terms.

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Step 3.1

Add $0$ and $0$ .

$2 (- 1 - \frac{\csc (x)}{2} - \frac{x^{2}}{4}) (0 + 0)$

Step 3.2

Add $0$ and $0$ .

$2 (- 1 - \frac{\csc (x)}{2} - \frac{x^{2}}{4}) \cdot 0$

Step 4

Simplify.

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Step 4.1

Apply the distributive property.

$(2 \cdot - 1 + 2 (- \frac{\csc (x)}{2}) + 2 (- \frac{x^{2}}{4})) \cdot 0$

Step 4.2

Apply the distributive property.

$2 \cdot - 1 \cdot 0 + 2 (- \frac{\csc (x)}{2}) \cdot 0 + 2 (- \frac{x^{2}}{4}) \cdot 0$

Step 4.3

Combine terms.

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Step 4.3.1

Multiply $2$ by $- 1$ .

$- 2 \cdot 0 + 2 (- \frac{\csc (x)}{2}) \cdot 0 + 2 (- \frac{x^{2}}{4}) \cdot 0$

Step 4.3.2

Multiply $- 2$ by $0$ .

$0 + 2 (- \frac{\csc (x)}{2}) \cdot 0 + 2 (- \frac{x^{2}}{4}) \cdot 0$

Step 4.3.3

Multiply $- 1$ by $2$ .

$0 - 2 \frac{\csc (x)}{2} \cdot 0 + 2 (- \frac{x^{2}}{4}) \cdot 0$

Step 4.3.4

Combine $- 2$ and $\frac{\csc (x)}{2}$ .

$0 + \frac{- 2 \csc (x)}{2} \cdot 0 + 2 (- \frac{x^{2}}{4}) \cdot 0$

Step 4.3.5

Cancel the common factor of $- 2$ and $2$ .

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Step 4.3.5.1

Factor $2$ out of $- 2 \csc (x)$ .

$0 + \frac{2 (- \csc (x))}{2} \cdot 0 + 2 (- \frac{x^{2}}{4}) \cdot 0$

Step 4.3.5.2

Cancel the common factors.

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Step 4.3.5.2.1

Factor $2$ out of $2$ .

$0 + \frac{2 (- \csc (x))}{2 (1)} \cdot 0 + 2 (- \frac{x^{2}}{4}) \cdot 0$

Step 4.3.5.2.2

Cancel the common factor.

$0 + \frac{2 (- \csc (x))}{2 \cdot 1} \cdot 0 + 2 (- \frac{x^{2}}{4}) \cdot 0$

Step 4.3.5.2.3

Rewrite the expression.

$0 + \frac{- \csc (x)}{1} \cdot 0 + 2 (- \frac{x^{2}}{4}) \cdot 0$

Step 4.3.5.2.4

Divide $- \csc (x)$ by $1$ .

$0 - \csc (x) \cdot 0 + 2 (- \frac{x^{2}}{4}) \cdot 0$

Step 4.3.6

Multiply $0$ by $- 1$ .

$0 + 0 \csc (x) + 2 (- \frac{x^{2}}{4}) \cdot 0$

Step 4.3.7

Multiply $0$ by $\csc (x)$ .

$0 + 0 + 2 (- \frac{x^{2}}{4}) \cdot 0$

Step 4.3.8

Add $0$ and $0$ .

$0 + 2 (- \frac{x^{2}}{4}) \cdot 0$

Step 4.3.9

Multiply $- 1$ by $2$ .

$0 - 2 \frac{x^{2}}{4} \cdot 0$

Step 4.3.10

Combine $- 2$ and $\frac{x^{2}}{4}$ .

$0 + \frac{- 2 x^{2}}{4} \cdot 0$

Step 4.3.11

Cancel the common factor of $- 2$ and $4$ .

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Step 4.3.11.1

Factor $2$ out of $- 2 x^{2}$ .

$0 + \frac{2 (- x^{2})}{4} \cdot 0$

Step 4.3.11.2

Cancel the common factors.

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Step 4.3.11.2.1

Factor $2$ out of $4$ .

$0 + \frac{2 (- x^{2})}{2 (2)} \cdot 0$

Step 4.3.11.2.2

Cancel the common factor.

$0 + \frac{2 (- x^{2})}{2 \cdot 2} \cdot 0$

Step 4.3.11.2.3

Rewrite the expression.

$0 + \frac{- x^{2}}{2} \cdot 0$

Step 4.3.12

Move the negative in front of the fraction.

$0 - \frac{x^{2}}{2} \cdot 0$

Step 4.3.13

Multiply $0$ by $- 1$ .

$0 + 0 \frac{x^{2}}{2}$

Step 4.3.14

Multiply $0$ by $\frac{x^{2}}{2}$ .

$0 + 0$

Step 4.3.15

Add $0$ and $0$ .

$0$