Calculus Examples

$\frac{x^{2} - 3}{x - 2}$

Step 1

Find the first derivative.

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Step 1.1

Differentiate using the Quotient Rule which states that $\frac{d}{d x} [\frac{f (x)}{g (x)}]$ is $\frac{g (x) \frac{d}{d x} [f (x)] - f (x) \frac{d}{d x} [g (x)]}{{g (x)}^{2}}$ where $f (x) = x^{2} - 3$ and $g (x) = x - 2$ .

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

Step 1.2

Differentiate.

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Step 1.2.1

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

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

Step 1.2.2

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

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

Step 1.2.3

Since $- 3$ is constant with respect to $x$ , the derivative of $- 3$ with respect to $x$ is $0$ .

$\frac{(x - 2) (2 x + 0) - (x^{2} - 3) \frac{d}{d x} [x - 2]}{{(x - 2)}^{2}}$

Step 1.2.4

Simplify the expression.

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Step 1.2.4.1

Add $2 x$ and $0$ .

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

Step 1.2.4.2

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

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

Step 1.2.5

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

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

Step 1.2.6

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

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

Step 1.2.7

Since $- 2$ is constant with respect to $x$ , the derivative of $- 2$ with respect to $x$ is $0$ .

$\frac{2 (x - 2) x - (x^{2} - 3) (1 + 0)}{{(x - 2)}^{2}}$

Step 1.2.8

Simplify the expression.

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Step 1.2.8.1

Add $1$ and $0$ .

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

Step 1.2.8.2

Multiply $- 1$ by $1$ .

$\frac{2 (x - 2) x - (x^{2} - 3)}{{(x - 2)}^{2}}$

Step 1.3

Simplify.

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Step 1.3.1

Apply the distributive property.

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

Step 1.3.2

Apply the distributive property.

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

Step 1.3.3

Apply the distributive property.

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

Step 1.3.4

Simplify the numerator.

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Step 1.3.4.1

Simplify each term.

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Step 1.3.4.1.1

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

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Step 1.3.4.1.1.1

Move $x$ .

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

Step 1.3.4.1.1.2

Multiply $x$ by $x$ .

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

Step 1.3.4.1.2

Multiply $2$ by $- 2$ .

$\frac{2 x^{2} - 4 x - x^{2} - - 3}{{(x - 2)}^{2}}$

Step 1.3.4.1.3

Multiply $- 1$ by $- 3$ .

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

Step 1.3.4.2

Subtract $x^{2}$ from $2 x^{2}$ .

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

Step 1.3.5

Factor $x^{2} - 4 x + 3$ using the AC method.

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Step 1.3.5.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $3$ and whose sum is $- 4$ .

$- 3, - 1$

Step 1.3.5.2

Write the factored form using these integers.

$f' (x) = \frac{(x - 3) (x - 1)}{{(x - 2)}^{2}}$

Step 2

Find the second derivative.

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Step 2.1

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

Step 2.2

Multiply the exponents in ${({(x - 2)}^{2})}^{2}$ .

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Step 2.2.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Step 2.2.2

Multiply $2$ by $2$ .

$\frac{{(x - 2)}^{2} \frac{d}{d x} [(x - 3) (x - 1)] - (x - 3) (x - 1) \frac{d}{d x} [{(x - 2)}^{2}]}{{(x - 2)}^{4}}$

Step 2.3

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 $f (x) = x - 3$ and $g (x) = x - 1$ .

$\frac{{(x - 2)}^{2} ((x - 3) \frac{d}{d x} [x - 1] + (x - 1) \frac{d}{d x} [x - 3]) - (x - 3) (x - 1) \frac{d}{d x} [{(x - 2)}^{2}]}{{(x - 2)}^{4}}$

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 $\frac{d}{d x} [x] + \frac{d}{d x} [- 1]$ .

$\frac{{(x - 2)}^{2} ((x - 3) (\frac{d}{d x} [x] + \frac{d}{d x} [- 1]) + (x - 1) \frac{d}{d x} [x - 3]) - (x - 3) (x - 1) \frac{d}{d x} [{(x - 2)}^{2}]}{{(x - 2)}^{4}}$

Step 2.4.2

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

$\frac{{(x - 2)}^{2} ((x - 3) (1 + \frac{d}{d x} [- 1]) + (x - 1) \frac{d}{d x} [x - 3]) - (x - 3) (x - 1) \frac{d}{d x} [{(x - 2)}^{2}]}{{(x - 2)}^{4}}$

Step 2.4.3

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

$\frac{{(x - 2)}^{2} ((x - 3) (1 + 0) + (x - 1) \frac{d}{d x} [x - 3]) - (x - 3) (x - 1) \frac{d}{d x} [{(x - 2)}^{2}]}{{(x - 2)}^{4}}$

Step 2.4.4

Simplify the expression.

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Step 2.4.4.1

Add $1$ and $0$ .

$\frac{{(x - 2)}^{2} ((x - 3) \cdot 1 + (x - 1) \frac{d}{d x} [x - 3]) - (x - 3) (x - 1) \frac{d}{d x} [{(x - 2)}^{2}]}{{(x - 2)}^{4}}$

Step 2.4.4.2

Multiply $x - 3$ by $1$ .

$\frac{{(x - 2)}^{2} (x - 3 + (x - 1) \frac{d}{d x} [x - 3]) - (x - 3) (x - 1) \frac{d}{d x} [{(x - 2)}^{2}]}{{(x - 2)}^{4}}$

Step 2.4.5

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

$\frac{{(x - 2)}^{2} (x - 3 + (x - 1) (\frac{d}{d x} [x] + \frac{d}{d x} [- 3])) - (x - 3) (x - 1) \frac{d}{d x} [{(x - 2)}^{2}]}{{(x - 2)}^{4}}$

Step 2.4.6

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

$\frac{{(x - 2)}^{2} (x - 3 + (x - 1) (1 + \frac{d}{d x} [- 3])) - (x - 3) (x - 1) \frac{d}{d x} [{(x - 2)}^{2}]}{{(x - 2)}^{4}}$

Step 2.4.7

Since $- 3$ is constant with respect to $x$ , the derivative of $- 3$ with respect to $x$ is $0$ .

$\frac{{(x - 2)}^{2} (x - 3 + (x - 1) (1 + 0)) - (x - 3) (x - 1) \frac{d}{d x} [{(x - 2)}^{2}]}{{(x - 2)}^{4}}$

Step 2.4.8

Simplify by adding terms.

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Step 2.4.8.1

Add $1$ and $0$ .

$\frac{{(x - 2)}^{2} (x - 3 + (x - 1) \cdot 1) - (x - 3) (x - 1) \frac{d}{d x} [{(x - 2)}^{2}]}{{(x - 2)}^{4}}$

Step 2.4.8.2

Multiply $x - 1$ by $1$ .

$\frac{{(x - 2)}^{2} (x - 3 + x - 1) - (x - 3) (x - 1) \frac{d}{d x} [{(x - 2)}^{2}]}{{(x - 2)}^{4}}$

Step 2.4.8.3

Add $x$ and $x$ .

$\frac{{(x - 2)}^{2} (2 x - 3 - 1) - (x - 3) (x - 1) \frac{d}{d x} [{(x - 2)}^{2}]}{{(x - 2)}^{4}}$

Step 2.4.8.4

Subtract $1$ from $- 3$ .

$\frac{{(x - 2)}^{2} (2 x - 4) - (x - 3) (x - 1) \frac{d}{d x} [{(x - 2)}^{2}]}{{(x - 2)}^{4}}$

Step 2.5

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

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Step 2.5.1

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

$\frac{{(x - 2)}^{2} (2 x - 4) - (x - 3) (x - 1) (\frac{d}{d u} [u^{2}] \frac{d}{d x} [x - 2])}{{(x - 2)}^{4}}$

Step 2.5.2

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

$\frac{{(x - 2)}^{2} (2 x - 4) - (x - 3) (x - 1) (2 u \frac{d}{d x} [x - 2])}{{(x - 2)}^{4}}$

Step 2.5.3

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

$\frac{{(x - 2)}^{2} (2 x - 4) - (x - 3) (x - 1) (2 (x - 2) \frac{d}{d x} [x - 2])}{{(x - 2)}^{4}}$

Step 2.6

Simplify with factoring out.

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Step 2.6.1

Multiply $2$ by $- 1$ .

$\frac{{(x - 2)}^{2} (2 x - 4) - 2 (x - 3) (x - 1) ((x - 2) \frac{d}{d x} [x - 2])}{{(x - 2)}^{4}}$

Step 2.6.2

Factor $x - 2$ out of ${(x - 2)}^{2} (2 x - 4) - 2 (x - 3) (x - 1) ((x - 2) \frac{d}{d x} [x - 2])$ .

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Step 2.6.2.1

Factor $x - 2$ out of ${(x - 2)}^{2} (2 x - 4)$ .

$\frac{(x - 2) ((x - 2) (2 x - 4)) - 2 (x - 3) (x - 1) ((x - 2) \frac{d}{d x} [x - 2])}{{(x - 2)}^{4}}$

Step 2.6.2.2

Factor $x - 2$ out of $- 2 (x - 3) (x - 1) ((x - 2) \frac{d}{d x} [x - 2])$ .

$\frac{(x - 2) ((x - 2) (2 x - 4)) + (x - 2) (- 2 (x - 3) (x - 1) (\frac{d}{d x} [x - 2]))}{{(x - 2)}^{4}}$

Step 2.6.2.3

Factor $x - 2$ out of $(x - 2) ((x - 2) (2 x - 4)) + (x - 2) (- 2 (x - 3) (x - 1) (\frac{d}{d x} [x - 2]))$ .

$\frac{(x - 2) ((x - 2) (2 x - 4) - 2 (x - 3) (x - 1) (\frac{d}{d x} [x - 2]))}{{(x - 2)}^{4}}$

Step 2.7

Cancel the common factors.

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Step 2.7.1

Factor $x - 2$ out of ${(x - 2)}^{4}$ .

$\frac{(x - 2) ((x - 2) (2 x - 4) - 2 (x - 3) (x - 1) (\frac{d}{d x} [x - 2]))}{(x - 2) {(x - 2)}^{3}}$

Step 2.7.2

Cancel the common factor.

$\frac{(x - 2) ((x - 2) (2 x - 4) - 2 (x - 3) (x - 1) (\frac{d}{d x} [x - 2]))}{(x - 2) {(x - 2)}^{3}}$

Step 2.7.3

Rewrite the expression.

$\frac{(x - 2) (2 x - 4) - 2 (x - 3) (x - 1) (\frac{d}{d x} [x - 2])}{{(x - 2)}^{3}}$

Step 2.8

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

$\frac{(x - 2) (2 x - 4) - 2 (x - 3) (x - 1) (\frac{d}{d x} [x] + \frac{d}{d x} [- 2])}{{(x - 2)}^{3}}$

Step 2.9

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

$\frac{(x - 2) (2 x - 4) - 2 (x - 3) (x - 1) (1 + \frac{d}{d x} [- 2])}{{(x - 2)}^{3}}$

Step 2.10

Since $- 2$ is constant with respect to $x$ , the derivative of $- 2$ with respect to $x$ is $0$ .

$\frac{(x - 2) (2 x - 4) - 2 (x - 3) (x - 1) (1 + 0)}{{(x - 2)}^{3}}$

Step 2.11

Simplify the expression.

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Step 2.11.1

Add $1$ and $0$ .

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

Step 2.11.2

Multiply $- 2$ by $1$ .

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

Step 2.12

Simplify.

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Step 2.12.1

Apply the distributive property.

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

Step 2.12.2

Simplify the numerator.

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Step 2.12.2.1

Simplify each term.

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Step 2.12.2.1.1

Expand $(x - 2) (2 x - 4)$ using the FOIL Method.

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Step 2.12.2.1.1.1

Apply the distributive property.

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

Step 2.12.2.1.1.2

Apply the distributive property.

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

Step 2.12.2.1.1.3

Apply the distributive property.

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

Step 2.12.2.1.2

Simplify and combine like terms.

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Step 2.12.2.1.2.1

Simplify each term.

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Step 2.12.2.1.2.1.1

Rewrite using the commutative property of multiplication.

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

Step 2.12.2.1.2.1.2

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

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Step 2.12.2.1.2.1.2.1

Move $x$ .

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

Step 2.12.2.1.2.1.2.2

Multiply $x$ by $x$ .

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

Step 2.12.2.1.2.1.3

Move $- 4$ to the left of $x$ .

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

Step 2.12.2.1.2.1.4

Multiply $2$ by $- 2$ .

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

Step 2.12.2.1.2.1.5

Multiply $- 2$ by $- 4$ .

$\frac{2 x^{2} - 4 x - 4 x + 8 + (- 2 x - 2 \cdot - 3) (x - 1)}{{(x - 2)}^{3}}$

Step 2.12.2.1.2.2

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

$\frac{2 x^{2} - 8 x + 8 + (- 2 x - 2 \cdot - 3) (x - 1)}{{(x - 2)}^{3}}$

Step 2.12.2.1.3

Multiply $- 2$ by $- 3$ .

$\frac{2 x^{2} - 8 x + 8 + (- 2 x + 6) (x - 1)}{{(x - 2)}^{3}}$

Step 2.12.2.1.4

Expand $(- 2 x + 6) (x - 1)$ using the FOIL Method.

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Step 2.12.2.1.4.1

Apply the distributive property.

$\frac{2 x^{2} - 8 x + 8 - 2 x (x - 1) + 6 (x - 1)}{{(x - 2)}^{3}}$

Step 2.12.2.1.4.2

Apply the distributive property.

$\frac{2 x^{2} - 8 x + 8 - 2 x \cdot x - 2 x \cdot - 1 + 6 (x - 1)}{{(x - 2)}^{3}}$

Step 2.12.2.1.4.3

Apply the distributive property.

$\frac{2 x^{2} - 8 x + 8 - 2 x \cdot x - 2 x \cdot - 1 + 6 x + 6 \cdot - 1}{{(x - 2)}^{3}}$

Step 2.12.2.1.5

Simplify and combine like terms.

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Step 2.12.2.1.5.1

Simplify each term.

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Step 2.12.2.1.5.1.1

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

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Step 2.12.2.1.5.1.1.1

Move $x$ .

$\frac{2 x^{2} - 8 x + 8 - 2 (x \cdot x) - 2 x \cdot - 1 + 6 x + 6 \cdot - 1}{{(x - 2)}^{3}}$

Step 2.12.2.1.5.1.1.2

Multiply $x$ by $x$ .

$\frac{2 x^{2} - 8 x + 8 - 2 x^{2} - 2 x \cdot - 1 + 6 x + 6 \cdot - 1}{{(x - 2)}^{3}}$

Step 2.12.2.1.5.1.2

Multiply $- 1$ by $- 2$ .

$\frac{2 x^{2} - 8 x + 8 - 2 x^{2} + 2 x + 6 x + 6 \cdot - 1}{{(x - 2)}^{3}}$

Step 2.12.2.1.5.1.3

Multiply $6$ by $- 1$ .

$\frac{2 x^{2} - 8 x + 8 - 2 x^{2} + 2 x + 6 x - 6}{{(x - 2)}^{3}}$

Step 2.12.2.1.5.2

Add $2 x$ and $6 x$ .

$\frac{2 x^{2} - 8 x + 8 - 2 x^{2} + 8 x - 6}{{(x - 2)}^{3}}$

Step 2.12.2.2

Combine the opposite terms in $2 x^{2} - 8 x + 8 - 2 x^{2} + 8 x - 6$ .

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Step 2.12.2.2.1

Subtract $2 x^{2}$ from $2 x^{2}$ .

$\frac{- 8 x + 8 + 0 + 8 x - 6}{{(x - 2)}^{3}}$

Step 2.12.2.2.2

Add $- 8 x + 8$ and $0$ .

$\frac{- 8 x + 8 + 8 x - 6}{{(x - 2)}^{3}}$

Step 2.12.2.2.3

Add $- 8 x$ and $8 x$ .

$\frac{0 + 8 - 6}{{(x - 2)}^{3}}$

Step 2.12.2.2.4

Add $0$ and $8$ .

$\frac{8 - 6}{{(x - 2)}^{3}}$

Step 2.12.2.3

Subtract $6$ from $8$ .

$f'' (x) = \frac{2}{{(x - 2)}^{3}}$