Calculus Examples

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

Step 1

Rewrite ${(1 - x)}^{2}$ as $(1 - x) (1 - x)$ .

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

Step 2

Expand $(1 - x) (1 - x)$ using the FOIL Method.

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Step 2.1

Apply the distributive property.

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

Step 2.2

Apply the distributive property.

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

Step 2.3

Apply the distributive property.

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

Step 3

Simplify and combine like terms.

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Step 3.1

Simplify each term.

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Step 3.1.1

Multiply $1$ by $1$ .

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

Step 3.1.2

Multiply $- x$ by $1$ .

$\frac{d}{d x} [(1 - x - x \cdot 1 - x (- x)) \sinh (2 x)]$

Step 3.1.3

Multiply $- 1$ by $1$ .

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

Step 3.1.4

Rewrite using the commutative property of multiplication.

$\frac{d}{d x} [(1 - x - x - 1 \cdot - 1 x \cdot x) \sinh (2 x)]$

Step 3.1.5

Multiply $x$ by $x$ by adding the exponents.

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Step 3.1.5.1

Move $x$ .

$\frac{d}{d x} [(1 - x - x - 1 \cdot - 1 (x \cdot x)) \sinh (2 x)]$

Step 3.1.5.2

Multiply $x$ by $x$ .

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

Step 3.1.6

Multiply $- 1$ by $- 1$ .

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

Step 3.1.7

Multiply $x^{2}$ by $1$ .

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

Step 3.2

Subtract $x$ from $- x$ .

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

Step 4

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 $f (x) = 1 - 2 x + x^{2}$ and $g (x) = \sinh (2 x)$ .

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

Step 5

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

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Step 5.1

To apply the Chain Rule, set $u$ as $2 x$ .

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

Step 5.2

The derivative of $\sinh (u)$ with respect to $u$ is $\cosh (u)$ .

$(1 - 2 x + x^{2}) (\cosh (u) \frac{d}{d x} [2 x]) + \sinh (2 x) \frac{d}{d x} [1 - 2 x + x^{2}]$

Step 5.3

Replace all occurrences of $u$ with $2 x$ .

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

Step 6

Differentiate.

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Step 6.1

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

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

Step 6.2

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

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

Step 6.3

Simplify the expression.

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Step 6.3.1

Multiply $2$ by $1$ .

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

Step 6.3.2

Move $2$ to the left of $(1 - 2 x + x^{2}) \cosh (2 x)$ .

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

Step 6.4

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

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

Step 6.5

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

$2 (1 - 2 x + x^{2}) \cosh (2 x) + \sinh (2 x) (0 + \frac{d}{d x} [- 2 x] + \frac{d}{d x} [x^{2}])$

Step 6.6

Add $0$ and $\frac{d}{d x} [- 2 x]$ .

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

Step 6.7

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

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

Step 6.8

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

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

Step 6.9

Multiply $- 2$ by $1$ .

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

Step 6.10

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

$2 (1 - 2 x + x^{2}) \cosh (2 x) + \sinh (2 x) (- 2 + 2 x)$

Step 7

Simplify.

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Step 7.1

Apply the distributive property.

$(2 \cdot 1 + 2 (- 2 x) + 2 x^{2}) \cosh (2 x) + \sinh (2 x) (- 2 + 2 x)$

Step 7.2

Apply the distributive property.

$2 \cdot 1 \cosh (2 x) + 2 (- 2 x) \cosh (2 x) + 2 x^{2} \cosh (2 x) + \sinh (2 x) (- 2 + 2 x)$

Step 7.3

Apply the distributive property.

$2 \cdot 1 \cosh (2 x) + 2 (- 2 x) \cosh (2 x) + 2 x^{2} \cosh (2 x) + \sinh (2 x) \cdot - 2 + \sinh (2 x) (2 x)$

Step 7.4

Combine terms.

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Step 7.4.1

Multiply $2$ by $1$ .

$2 \cosh (2 x) + 2 (- 2 x) \cosh (2 x) + 2 x^{2} \cosh (2 x) + \sinh (2 x) \cdot - 2 + \sinh (2 x) (2 x)$

Step 7.4.2

Multiply $- 2$ by $2$ .

$2 \cosh (2 x) - 4 x \cosh (2 x) + 2 x^{2} \cosh (2 x) + \sinh (2 x) \cdot - 2 + \sinh (2 x) (2 x)$

Step 7.4.3

Move $- 2$ to the left of $\sinh (2 x)$ .

$2 \cosh (2 x) - 4 x \cosh (2 x) + 2 x^{2} \cosh (2 x) - 2 \sinh (2 x) + \sinh (2 x) (2 x)$

Step 7.5

Reorder terms.

$- 4 x \cosh (2 x) + 2 x^{2} \cosh (2 x) + 2 x \sinh (2 x) + 2 \cosh (2 x) - 2 \sinh (2 x)$