Calculus Examples

$y = (2 x - \csc^{2} (3 x - 1))$

Step 1

Remove parentheses.

$y = 2 x - \csc^{2} (3 x - 1)$

Step 2

Differentiate both sides of the equation.

$\frac{d}{d x} (y) = \frac{d}{d x} (2 x - \csc^{2} (3 x - 1))$

Step 3

The derivative of $y$ with respect to $x$ is $y'$ .

$y'$

Step 4

Differentiate the right side of the equation.

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Step 4.1

By the Sum Rule, the derivative of $2 x - \csc^{2} (3 x - 1)$ with respect to $x$ is $\frac{d}{d x} [2 x] + \frac{d}{d x} [- \csc^{2} (3 x - 1)]$ .

$\frac{d}{d x} [2 x] + \frac{d}{d x} [- \csc^{2} (3 x - 1)]$

Step 4.2

Evaluate $\frac{d}{d x} [2 x]$ .

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Step 4.2.1

Since $2$ is constant with respect to $x$ , the derivative of $2 x$ with respect to $x$ is $2 \frac{d}{d x} [x]$ .

$2 \frac{d}{d x} [x] + \frac{d}{d x} [- \csc^{2} (3 x - 1)]$

Step 4.2.2

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

$2 \cdot 1 + \frac{d}{d x} [- \csc^{2} (3 x - 1)]$

Step 4.2.3

Multiply $2$ by $1$ .

$2 + \frac{d}{d x} [- \csc^{2} (3 x - 1)]$

Step 4.3

Evaluate $\frac{d}{d x} [- \csc^{2} (3 x - 1)]$ .

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Step 4.3.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- \csc^{2} (3 x - 1)$ with respect to $x$ is $- \frac{d}{d x} [\csc^{2} (3 x - 1)]$ .

$2 - \frac{d}{d x} [\csc^{2} (3 x - 1)]$

Step 4.3.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{2}$ and $g (x) = \csc (3 x - 1)$ .

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Step 4.3.2.1

To apply the Chain Rule, set $u_{1}$ as $\csc (3 x - 1)$ .

$2 - (\frac{d}{d u_{1}} [{u_{1}}^{2}] \frac{d}{d x} [\csc (3 x - 1)])$

Step 4.3.2.2

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

$2 - (2 u_{1} \frac{d}{d x} [\csc (3 x - 1)])$

Step 4.3.2.3

Replace all occurrences of $u_{1}$ with $\csc (3 x - 1)$ .

$2 - (2 \csc (3 x - 1) \frac{d}{d x} [\csc (3 x - 1)])$

Step 4.3.3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \csc (x)$ and $g (x) = 3 x - 1$ .

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Step 4.3.3.1

To apply the Chain Rule, set $u_{2}$ as $3 x - 1$ .

$2 - (2 \csc (3 x - 1) (\frac{d}{d u_{2}} [\csc (u_{2})] \frac{d}{d x} [3 x - 1]))$

Step 4.3.3.2

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

$2 - (2 \csc (3 x - 1) (- \csc (u_{2}) \cot (u_{2}) \frac{d}{d x} [3 x - 1]))$

Step 4.3.3.3

Replace all occurrences of $u_{2}$ with $3 x - 1$ .

$2 - (2 \csc (3 x - 1) (- \csc (3 x - 1) \cot (3 x - 1) \frac{d}{d x} [3 x - 1]))$

Step 4.3.4

By the Sum Rule, the derivative of $3 x - 1$ with respect to $x$ is $\frac{d}{d x} [3 x] + \frac{d}{d x} [- 1]$ .

$2 - (2 \csc (3 x - 1) (- \csc (3 x - 1) \cot (3 x - 1) (\frac{d}{d x} [3 x] + \frac{d}{d x} [- 1])))$

Step 4.3.5

Since $3$ is constant with respect to $x$ , the derivative of $3 x$ with respect to $x$ is $3 \frac{d}{d x} [x]$ .

$2 - (2 \csc (3 x - 1) (- \csc (3 x - 1) \cot (3 x - 1) (3 \frac{d}{d x} [x] + \frac{d}{d x} [- 1])))$

Step 4.3.6

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

$2 - (2 \csc (3 x - 1) (- \csc (3 x - 1) \cot (3 x - 1) (3 \cdot 1 + \frac{d}{d x} [- 1])))$

Step 4.3.7

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

$2 - (2 \csc (3 x - 1) (- \csc (3 x - 1) \cot (3 x - 1) (3 \cdot 1 + 0)))$

Step 4.3.8

Multiply $3$ by $1$ .

$2 - (2 \csc (3 x - 1) (- \csc (3 x - 1) \cot (3 x - 1) (3 + 0)))$

Step 4.3.9

Add $3$ and $0$ .

$2 - (2 \csc (3 x - 1) (- \csc (3 x - 1) \cot (3 x - 1) \cdot 3))$

Step 4.3.10

Multiply $3$ by $- 1$ .

$2 - (2 \csc (3 x - 1) (- 3 \csc (3 x - 1) \cot (3 x - 1)))$

Step 4.3.11

Multiply $- 3$ by $2$ .

$2 - (- 6 \csc (3 x - 1) (\csc (3 x - 1) \cot (3 x - 1)))$

Step 4.3.12

Raise $\csc (3 x - 1)$ to the power of $1$ .

$2 - (- 6 (\csc^{1} (3 x - 1) \csc (3 x - 1)) \cot (3 x - 1))$

Step 4.3.13

Raise $\csc (3 x - 1)$ to the power of $1$ .

$2 - (- 6 (\csc^{1} (3 x - 1) \csc^{1} (3 x - 1)) \cot (3 x - 1))$

Step 4.3.14

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$2 - (- 6 {\csc (3 x - 1)}^{1 + 1} \cot (3 x - 1))$

Step 4.3.15

Add $1$ and $1$ .

$2 - (- 6 \csc^{2} (3 x - 1) \cot (3 x - 1))$

Step 4.3.16

Multiply $- 6$ by $- 1$ .

$2 + 6 \csc^{2} (3 x - 1) \cot (3 x - 1)$

Step 4.4

Reorder terms.

$6 \csc^{2} (3 x - 1) \cot (3 x - 1) + 2$

Step 5

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

$y' = 6 \csc^{2} (3 x - 1) \cot (3 x - 1) + 2$

Step 6

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

$\frac{d y}{d x} = 6 \csc^{2} (3 x - 1) \cot (3 x - 1) + 2$