Calculus Examples

$f (x) = {({(x - 1)}^{2} - x)}^{2}$ at $x = 2$

Step 1

Find the corresponding $y$ -value to $x = 2$ .

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Step 1.1

Substitute $2$ in for $x$ .

$y = {({((2) - 1)}^{2} - (2))}^{2}$

Step 1.2

Solve for $y$ .

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Step 1.2.1

Remove parentheses.

$y = {({(2 - 1)}^{2} - (2))}^{2}$

Step 1.2.2

Remove parentheses.

$y = {({((2) - 1)}^{2} - (2))}^{2}$

Step 1.2.3

Simplify ${({((2) - 1)}^{2} - (2))}^{2}$ .

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Step 1.2.3.1

Simplify each term.

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Step 1.2.3.1.1

Subtract $1$ from $2$ .

$y = {(1^{2} - (2))}^{2}$

Step 1.2.3.1.2

One to any power is one.

$y = {(1 - (2))}^{2}$

Step 1.2.3.1.3

Multiply $- 1$ by $2$ .

$y = {(1 - 2)}^{2}$

Step 1.2.3.2

Simplify the expression.

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Step 1.2.3.2.1

Subtract $2$ from $1$ .

$y = {(- 1)}^{2}$

Step 1.2.3.2.2

Raise $- 1$ to the power of $2$ .

$y = 1$

Step 2

Find the first derivative and evaluate at $x = 2$ and $y = 1$ to find the slope of the tangent line.

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Step 2.1

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

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Step 2.1.1

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

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

Step 2.1.2

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

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

Step 2.1.3

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

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

Step 2.2

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

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

Step 2.3

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

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Step 2.3.1

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

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

Step 2.3.2

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

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

Step 2.3.3

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

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

Step 2.4

Differentiate.

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Step 2.4.1

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

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

Step 2.4.2

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

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

Step 2.4.3

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

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

Step 2.4.4

Simplify the expression.

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Step 2.4.4.1

Add $1$ and $0$ .

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

Step 2.4.4.2

Multiply $2$ by $1$ .

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

Step 2.4.5

Since $- 1$ is constant with respect to $x$ , the derivative of $- x$ with respect to $x$ is $- \frac{d}{d x} [x]$ .

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

Step 2.4.6

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

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

Step 2.4.7

Multiply $- 1$ by $1$ .

$2 ({(x - 1)}^{2} - x) (2 (x - 1) - 1)$

Step 2.5

Simplify.

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Step 2.5.1

Apply the distributive property.

$(2 {(x - 1)}^{2} + 2 (- x)) (2 (x - 1) - 1)$

Step 2.5.2

Apply the distributive property.

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

Step 2.5.3

Combine terms.

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Step 2.5.3.1

Multiply $- 1$ by $2$ .

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

Step 2.5.3.2

Multiply $2$ by $- 1$ .

$(2 {(x - 1)}^{2} - 2 x) (2 x - 2 - 1)$

Step 2.5.3.3

Subtract $1$ from $- 2$ .

$(2 {(x - 1)}^{2} - 2 x) (2 x - 3)$

Step 2.5.4

Simplify each term.

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Step 2.5.4.1

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

$(2 ((x - 1) (x - 1)) - 2 x) (2 x - 3)$

Step 2.5.4.2

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

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Step 2.5.4.2.1

Apply the distributive property.

$(2 (x (x - 1) - 1 (x - 1)) - 2 x) (2 x - 3)$

Step 2.5.4.2.2

Apply the distributive property.

$(2 (x \cdot x + x \cdot - 1 - 1 (x - 1)) - 2 x) (2 x - 3)$

Step 2.5.4.2.3

Apply the distributive property.

$(2 (x \cdot x + x \cdot - 1 - 1 x - 1 \cdot - 1) - 2 x) (2 x - 3)$

Step 2.5.4.3

Simplify and combine like terms.

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Step 2.5.4.3.1

Simplify each term.

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Step 2.5.4.3.1.1

Multiply $x$ by $x$ .

$(2 (x^{2} + x \cdot - 1 - 1 x - 1 \cdot - 1) - 2 x) (2 x - 3)$

Step 2.5.4.3.1.2

Move $- 1$ to the left of $x$ .

$(2 (x^{2} - 1 \cdot x - 1 x - 1 \cdot - 1) - 2 x) (2 x - 3)$

Step 2.5.4.3.1.3

Rewrite $- 1 x$ as $- x$ .

$(2 (x^{2} - x - 1 x - 1 \cdot - 1) - 2 x) (2 x - 3)$

Step 2.5.4.3.1.4

Rewrite $- 1 x$ as $- x$ .

$(2 (x^{2} - x - x - 1 \cdot - 1) - 2 x) (2 x - 3)$

Step 2.5.4.3.1.5

Multiply $- 1$ by $- 1$ .

$(2 (x^{2} - x - x + 1) - 2 x) (2 x - 3)$

Step 2.5.4.3.2

Subtract $x$ from $- x$ .

$(2 (x^{2} - 2 x + 1) - 2 x) (2 x - 3)$

Step 2.5.4.4

Apply the distributive property.

$(2 x^{2} + 2 (- 2 x) + 2 \cdot 1 - 2 x) (2 x - 3)$

Step 2.5.4.5

Simplify.

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Step 2.5.4.5.1

Multiply $- 2$ by $2$ .

$(2 x^{2} - 4 x + 2 \cdot 1 - 2 x) (2 x - 3)$

Step 2.5.4.5.2

Multiply $2$ by $1$ .

$(2 x^{2} - 4 x + 2 - 2 x) (2 x - 3)$

Step 2.5.5

Subtract $2 x$ from $- 4 x$ .

$(2 x^{2} - 6 x + 2) (2 x - 3)$

Step 2.5.6

Expand $(2 x^{2} - 6 x + 2) (2 x - 3)$ by multiplying each term in the first expression by each term in the second expression.

$2 x^{2} (2 x) + 2 x^{2} \cdot - 3 - 6 x (2 x) - 6 x \cdot - 3 + 2 (2 x) + 2 \cdot - 3$

Step 2.5.7

Simplify each term.

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Step 2.5.7.1

Rewrite using the commutative property of multiplication.

$2 \cdot 2 x^{2} x + 2 x^{2} \cdot - 3 - 6 x (2 x) - 6 x \cdot - 3 + 2 (2 x) + 2 \cdot - 3$

Step 2.5.7.2

Multiply $x^{2}$ by $x$ by adding the exponents.

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Step 2.5.7.2.1

Move $x$ .

$2 \cdot 2 (x \cdot x^{2}) + 2 x^{2} \cdot - 3 - 6 x (2 x) - 6 x \cdot - 3 + 2 (2 x) + 2 \cdot - 3$

Step 2.5.7.2.2

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

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Step 2.5.7.2.2.1

Raise $x$ to the power of $1$ .

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

Step 2.5.7.2.2.2

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

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

Step 2.5.7.2.3

Add $1$ and $2$ .

$2 \cdot 2 x^{3} + 2 x^{2} \cdot - 3 - 6 x (2 x) - 6 x \cdot - 3 + 2 (2 x) + 2 \cdot - 3$

Step 2.5.7.3

Multiply $2$ by $2$ .

$4 x^{3} + 2 x^{2} \cdot - 3 - 6 x (2 x) - 6 x \cdot - 3 + 2 (2 x) + 2 \cdot - 3$

Step 2.5.7.4

Multiply $- 3$ by $2$ .

$4 x^{3} - 6 x^{2} - 6 x (2 x) - 6 x \cdot - 3 + 2 (2 x) + 2 \cdot - 3$

Step 2.5.7.5

Rewrite using the commutative property of multiplication.

$4 x^{3} - 6 x^{2} - 6 \cdot 2 x \cdot x - 6 x \cdot - 3 + 2 (2 x) + 2 \cdot - 3$

Step 2.5.7.6

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

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Step 2.5.7.6.1

Move $x$ .

$4 x^{3} - 6 x^{2} - 6 \cdot 2 (x \cdot x) - 6 x \cdot - 3 + 2 (2 x) + 2 \cdot - 3$

Step 2.5.7.6.2

Multiply $x$ by $x$ .

$4 x^{3} - 6 x^{2} - 6 \cdot 2 x^{2} - 6 x \cdot - 3 + 2 (2 x) + 2 \cdot - 3$

Step 2.5.7.7

Multiply $- 6$ by $2$ .

$4 x^{3} - 6 x^{2} - 12 x^{2} - 6 x \cdot - 3 + 2 (2 x) + 2 \cdot - 3$

Step 2.5.7.8

Multiply $- 3$ by $- 6$ .

$4 x^{3} - 6 x^{2} - 12 x^{2} + 18 x + 2 (2 x) + 2 \cdot - 3$

Step 2.5.7.9

Multiply $2$ by $2$ .

$4 x^{3} - 6 x^{2} - 12 x^{2} + 18 x + 4 x + 2 \cdot - 3$

Step 2.5.7.10

Multiply $2$ by $- 3$ .

$4 x^{3} - 6 x^{2} - 12 x^{2} + 18 x + 4 x - 6$

Step 2.5.8

Subtract $12 x^{2}$ from $- 6 x^{2}$ .

$4 x^{3} - 18 x^{2} + 18 x + 4 x - 6$

Step 2.5.9

Add $18 x$ and $4 x$ .

$4 x^{3} - 18 x^{2} + 22 x - 6$

Step 2.6

Evaluate the derivative at $x = 2$ .

$4 {(2)}^{3} - 18 {(2)}^{2} + 22 (2) - 6$

Step 2.7

Simplify.

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Step 2.7.1

Simplify each term.

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Step 2.7.1.1

Raise $2$ to the power of $3$ .

$4 \cdot 8 - 18 {(2)}^{2} + 22 (2) - 6$

Step 2.7.1.2

Multiply $4$ by $8$ .

$32 - 18 {(2)}^{2} + 22 (2) - 6$

Step 2.7.1.3

Raise $2$ to the power of $2$ .

$32 - 18 \cdot 4 + 22 (2) - 6$

Step 2.7.1.4

Multiply $- 18$ by $4$ .

$32 - 72 + 22 (2) - 6$

Step 2.7.1.5

Multiply $22$ by $2$ .

$32 - 72 + 44 - 6$

Step 2.7.2

Simplify by adding and subtracting.

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Step 2.7.2.1

Subtract $72$ from $32$ .

$- 40 + 44 - 6$

Step 2.7.2.2

Add $- 40$ and $44$ .

$4 - 6$

Step 2.7.2.3

Subtract $6$ from $4$ .

$- 2$

Step 3

Plug the slope and point values into the point-slope formula and solve for $y$ .

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Step 3.1

Use the slope $- 2$ and a given point $(2, 1)$ to substitute for $x_{1}$ and $y_{1}$ in the point-slope form $y - y_{1} = m (x - x_{1})$ , which is derived from the slope equation $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (1) = - 2 \cdot (x - (2))$

Step 3.2

Simplify the equation and keep it in point-slope form.

$y - 1 = - 2 \cdot (x - 2)$

Step 3.3

Solve for $y$ .

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Step 3.3.1

Simplify $- 2 \cdot (x - 2)$ .

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Step 3.3.1.1

Rewrite.

$y - 1 = 0 + 0 - 2 \cdot (x - 2)$

Step 3.3.1.2

Simplify by adding zeros.

$y - 1 = - 2 \cdot (x - 2)$

Step 3.3.1.3

Apply the distributive property.

$y - 1 = - 2 x - 2 \cdot - 2$

Step 3.3.1.4

Multiply $- 2$ by $- 2$ .

$y - 1 = - 2 x + 4$

Step 3.3.2

Move all terms not containing $y$ to the right side of the equation.

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Step 3.3.2.1

Add $1$ to both sides of the equation.

$y = - 2 x + 4 + 1$

Step 3.3.2.2

Add $4$ and $1$ .

$y = - 2 x + 5$

Step 4