Calculus Examples

$y = \arctan (\sin (x^{2} - 6 x))$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (y) = \frac{d}{d x} (\arctan (\sin (x^{2} - 6 x)))$

Step 2

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

$y'$

Step 3

Differentiate the right side of the equation.

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Step 3.1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \arctan (x)$ and $g (x) = \sin (x^{2} - 6 x)$ .

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Step 3.1.1

To apply the Chain Rule, set $u_{1}$ as $\sin (x^{2} - 6 x)$ .

$\frac{d}{d u_{1}} [\arctan (u_{1})] \frac{d}{d x} [\sin (x^{2} - 6 x)]$

Step 3.1.2

The derivative of $\arctan (u_{1})$ with respect to $u_{1}$ is $\frac{1}{1 + {u_{1}}^{2}}$ .

$\frac{1}{1 + {u_{1}}^{2}} \frac{d}{d x} [\sin (x^{2} - 6 x)]$

Step 3.1.3

Replace all occurrences of $u_{1}$ with $\sin (x^{2} - 6 x)$ .

$\frac{1}{1 + \sin^{2} (x^{2} - 6 x)} \frac{d}{d x} [\sin (x^{2} - 6 x)]$

Step 3.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \sin (x)$ and $g (x) = x^{2} - 6 x$ .

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Step 3.2.1

To apply the Chain Rule, set $u_{2}$ as $x^{2} - 6 x$ .

$\frac{1}{1 + \sin^{2} (x^{2} - 6 x)} (\frac{d}{d u_{2}} [\sin (u_{2})] \frac{d}{d x} [x^{2} - 6 x])$

Step 3.2.2

The derivative of $\sin (u_{2})$ with respect to $u_{2}$ is $\cos (u_{2})$ .

$\frac{1}{1 + \sin^{2} (x^{2} - 6 x)} (\cos (u_{2}) \frac{d}{d x} [x^{2} - 6 x])$

Step 3.2.3

Replace all occurrences of $u_{2}$ with $x^{2} - 6 x$ .

$\frac{1}{1 + \sin^{2} (x^{2} - 6 x)} (\cos (x^{2} - 6 x) \frac{d}{d x} [x^{2} - 6 x])$

Step 3.3

Differentiate.

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Step 3.3.1

Combine $\cos (x^{2} - 6 x)$ and $\frac{1}{1 + \sin^{2} (x^{2} - 6 x)}$ .

$\frac{\cos (x^{2} - 6 x)}{1 + \sin^{2} (x^{2} - 6 x)} \frac{d}{d x} [x^{2} - 6 x]$

Step 3.3.2

By the Sum Rule, the derivative of $x^{2} - 6 x$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 6 x]$ .

$\frac{\cos (x^{2} - 6 x)}{1 + \sin^{2} (x^{2} - 6 x)} (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 6 x])$

Step 3.3.3

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

$\frac{\cos (x^{2} - 6 x)}{1 + \sin^{2} (x^{2} - 6 x)} (2 x + \frac{d}{d x} [- 6 x])$

Step 3.3.4

Since $- 6$ is constant with respect to $x$ , the derivative of $- 6 x$ with respect to $x$ is $- 6 \frac{d}{d x} [x]$ .

$\frac{\cos (x^{2} - 6 x)}{1 + \sin^{2} (x^{2} - 6 x)} (2 x - 6 \frac{d}{d x} [x])$

Step 3.3.5

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

$\frac{\cos (x^{2} - 6 x)}{1 + \sin^{2} (x^{2} - 6 x)} (2 x - 6 \cdot 1)$

Step 3.3.6

Multiply $- 6$ by $1$ .

$\frac{\cos (x^{2} - 6 x)}{1 + \sin^{2} (x^{2} - 6 x)} (2 x - 6)$

Step 3.4

Simplify.

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Step 3.4.1

Reorder the factors of $\frac{\cos (x^{2} - 6 x)}{1 + \sin^{2} (x^{2} - 6 x)} (2 x - 6)$ .

$(2 x - 6) \frac{\cos (x^{2} - 6 x)}{1 + \sin^{2} (x^{2} - 6 x)}$

Step 3.4.2

Apply the distributive property.

$2 x \frac{\cos (x^{2} - 6 x)}{1 + \sin^{2} (x^{2} - 6 x)} - 6 \frac{\cos (x^{2} - 6 x)}{1 + \sin^{2} (x^{2} - 6 x)}$

Step 3.4.3

Multiply $2 x \frac{\cos (x^{2} - 6 x)}{1 + \sin^{2} (x^{2} - 6 x)}$ .

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Step 3.4.3.1

Combine $2$ and $\frac{\cos (x^{2} - 6 x)}{1 + \sin^{2} (x^{2} - 6 x)}$ .

$x \frac{2 \cos (x^{2} - 6 x)}{1 + \sin^{2} (x^{2} - 6 x)} - 6 \frac{\cos (x^{2} - 6 x)}{1 + \sin^{2} (x^{2} - 6 x)}$

Step 3.4.3.2

Combine $x$ and $\frac{2 \cos (x^{2} - 6 x)}{1 + \sin^{2} (x^{2} - 6 x)}$ .

$\frac{x (2 \cos (x^{2} - 6 x))}{1 + \sin^{2} (x^{2} - 6 x)} - 6 \frac{\cos (x^{2} - 6 x)}{1 + \sin^{2} (x^{2} - 6 x)}$

Step 3.4.4

Combine $- 6$ and $\frac{\cos (x^{2} - 6 x)}{1 + \sin^{2} (x^{2} - 6 x)}$ .

$\frac{x (2 \cos (x^{2} - 6 x))}{1 + \sin^{2} (x^{2} - 6 x)} + \frac{- 6 \cos (x^{2} - 6 x)}{1 + \sin^{2} (x^{2} - 6 x)}$

Step 3.4.5

Simplify each term.

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Step 3.4.5.1

Move $2$ to the left of $x$ .

$\frac{2 \cdot x \cos (x^{2} - 6 x)}{1 + \sin^{2} (x^{2} - 6 x)} + \frac{- 6 \cos (x^{2} - 6 x)}{1 + \sin^{2} (x^{2} - 6 x)}$

Step 3.4.5.2

Move the negative in front of the fraction.

$\frac{2 x \cos (x^{2} - 6 x)}{1 + \sin^{2} (x^{2} - 6 x)} - \frac{6 \cos (x^{2} - 6 x)}{1 + \sin^{2} (x^{2} - 6 x)}$

Step 3.4.6

Combine the numerators over the common denominator.

$\frac{2 x \cos (x^{2} - 6 x) - 6 \cos (x^{2} - 6 x)}{1 + \sin^{2} (x^{2} - 6 x)}$

Step 3.4.7

Factor $2 \cos (x^{2} - 6 x)$ out of $2 x \cos (x^{2} - 6 x) - 6 \cos (x^{2} - 6 x)$ .

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Step 3.4.7.1

Factor $2 \cos (x^{2} - 6 x)$ out of $2 x \cos (x^{2} - 6 x)$ .

$\frac{2 \cos (x^{2} - 6 x) (x) - 6 \cos (x^{2} - 6 x)}{1 + \sin^{2} (x^{2} - 6 x)}$

Step 3.4.7.2

Factor $2 \cos (x^{2} - 6 x)$ out of $- 6 \cos (x^{2} - 6 x)$ .

$\frac{2 \cos (x^{2} - 6 x) (x) + 2 \cos (x^{2} - 6 x) (- 3)}{1 + \sin^{2} (x^{2} - 6 x)}$

Step 3.4.7.3

Factor $2 \cos (x^{2} - 6 x)$ out of $2 \cos (x^{2} - 6 x) (x) + 2 \cos (x^{2} - 6 x) (- 3)$ .

$\frac{2 \cos (x^{2} - 6 x) (x - 3)}{1 + \sin^{2} (x^{2} - 6 x)}$

Step 4

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

$y' = \frac{2 \cos (x^{2} - 6 x) (x - 3)}{1 + \sin^{2} (x^{2} - 6 x)}$

Step 5

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

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