Calculus Examples

$y = \sqrt{1 - 2 x t + t^{2}}$

Step 1

Find the first derivative.

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Step 1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{1 - 2 x t + t^{2}}$ as ${(1 - 2 x t + t^{2})}^{\frac{1}{2}}$ .

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

Step 1.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{\frac{1}{2}}$ and $g (x) = 1 - 2 x t + t^{2}$ .

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Step 1.2.1

To apply the Chain Rule, set $u$ as $1 - 2 x t + t^{2}$ .

$\frac{d}{d u} [u^{\frac{1}{2}}] \frac{d}{d x} [1 - 2 x t + t^{2}]$

Step 1.2.2

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

$\frac{1}{2} u^{\frac{1}{2} - 1} \frac{d}{d x} [1 - 2 x t + t^{2}]$

Step 1.2.3

Replace all occurrences of $u$ with $1 - 2 x t + t^{2}$ .

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

Step 1.3

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 1.4

Combine $- 1$ and $\frac{2}{2}$ .

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

Step 1.5

Combine the numerators over the common denominator.

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

Step 1.6

Simplify the numerator.

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Step 1.6.1

Multiply $- 1$ by $2$ .

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

Step 1.6.2

Subtract $2$ from $1$ .

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

Step 1.7

Combine fractions.

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Step 1.7.1

Move the negative in front of the fraction.

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

Step 1.7.2

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

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

Step 1.7.3

Move ${(1 - 2 x t + t^{2})}^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 1.8

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

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

Step 1.9

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

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

Step 1.10

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

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

Step 1.11

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

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

Step 1.12

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

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

Step 1.13

Multiply $- 2$ by $1$ .

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

Step 1.14

Since $t^{2}$ is constant with respect to $x$ , the derivative of $t^{2}$ with respect to $x$ is $0$ .

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

Step 1.15

Simplify terms.

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Step 1.15.1

Add $- 2 t$ and $0$ .

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

Step 1.15.2

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

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

Step 1.15.3

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

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

Step 1.15.4

Factor $2$ out of $- 2 t$ .

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

Step 1.16

Cancel the common factors.

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Step 1.16.1

Factor $2$ out of $2 {(1 - 2 x t + t^{2})}^{\frac{1}{2}}$ .

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

Step 1.16.2

Cancel the common factor.

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

Step 1.16.3

Rewrite the expression.

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

Step 1.17

Move the negative in front of the fraction.

$f' (x) = - \frac{t}{{(1 - 2 x t + t^{2})}^{\frac{1}{2}}}$

Step 2

Find the second derivative.

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Step 2.1

Differentiate using the Constant Multiple Rule.

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Step 2.1.1

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

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

Step 2.1.2

Apply basic rules of exponents.

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Step 2.1.2.1

Rewrite $\frac{1}{{(1 - 2 x t + t^{2})}^{\frac{1}{2}}}$ as ${({(1 - 2 x t + t^{2})}^{\frac{1}{2}})}^{- 1}$ .

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

Step 2.1.2.2

Multiply the exponents in ${({(1 - 2 x t + t^{2})}^{\frac{1}{2}})}^{- 1}$ .

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Step 2.1.2.2.1

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

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

Step 2.1.2.2.2

Combine $\frac{1}{2}$ and $- 1$ .

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

Step 2.1.2.2.3

Move the negative in front of the fraction.

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

Step 2.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{- \frac{1}{2}}$ and $g (x) = 1 - 2 x t + t^{2}$ .

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Step 2.2.1

To apply the Chain Rule, set $u$ as $1 - 2 x t + t^{2}$ .

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

Step 2.2.2

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

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

Step 2.2.3

Replace all occurrences of $u$ with $1 - 2 x t + t^{2}$ .

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

Step 2.3

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 2.4

Combine $- 1$ and $\frac{2}{2}$ .

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

Step 2.5

Combine the numerators over the common denominator.

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

Step 2.6

Simplify the numerator.

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Step 2.6.1

Multiply $- 1$ by $2$ .

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

Step 2.6.2

Subtract $2$ from $- 1$ .

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

Step 2.7

Combine fractions.

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Step 2.7.1

Move the negative in front of the fraction.

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

Step 2.7.2

Combine ${(1 - 2 x t + t^{2})}^{- \frac{3}{2}}$ and $\frac{1}{2}$ .

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

Step 2.7.3

Simplify the expression.

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Step 2.7.3.1

Move ${(1 - 2 x t + t^{2})}^{- \frac{3}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 2.7.3.2

Multiply $- 1$ by $- 1$ .

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

Step 2.7.3.3

Multiply $t$ by $1$ .

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

Step 2.7.4

Combine $\frac{1}{2 {(1 - 2 x t + t^{2})}^{\frac{3}{2}}}$ and $t$ .

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

Step 2.8

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

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

Step 2.9

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

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

Step 2.10

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

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

Step 2.11

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

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

Step 2.12

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

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

Step 2.13

Multiply $- 2$ by $1$ .

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

Step 2.14

Since $t^{2}$ is constant with respect to $x$ , the derivative of $t^{2}$ with respect to $x$ is $0$ .

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

Step 2.15

Combine fractions.

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Step 2.15.1

Add $- 2 t$ and $0$ .

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

Step 2.15.2

Combine $- 2$ and $\frac{t}{2 {(1 - 2 x t + t^{2})}^{\frac{3}{2}}}$ .

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

Step 2.15.3

Combine $\frac{- 2 t}{2 {(1 - 2 x t + t^{2})}^{\frac{3}{2}}}$ and $t$ .

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

Step 2.16

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

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

Step 2.17

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

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

Step 2.18

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

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

Step 2.19

Add $1$ and $1$ .

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

Step 2.20

Factor $2$ out of $- 2 t^{2}$ .

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

Step 2.21

Cancel the common factors.

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Step 2.21.1

Factor $2$ out of $2 {(1 - 2 x t + t^{2})}^{\frac{3}{2}}$ .

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

Step 2.21.2

Cancel the common factor.

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

Step 2.21.3

Rewrite the expression.

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

Step 2.22

Move the negative in front of the fraction.

$f'' (x) = - \frac{t^{2}}{{(1 - 2 x t + t^{2})}^{\frac{3}{2}}}$