Calculus Examples

$x^{n}$

Step 1

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

$f' (x) = n x^{n - 1}$

Step 2

Find the second derivative.

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Step 2.1

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

$n \frac{d}{d x} [x^{n - 1}]$

Step 2.2

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

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

Step 2.3

Simplify the expression.

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Step 2.3.1

Subtract $1$ from $- 1$ .

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

Step 2.3.2

Reorder the factors of $n ((n - 1) x^{n - 2})$ .

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

Step 3

Find the third derivative.

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

Subtract $1$ from $- 2$ .

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

Step 3.4

Simplify.

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Step 3.4.1

Apply the distributive property.

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

Step 3.4.2

Combine terms.

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Step 3.4.2.1

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

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

Step 3.4.2.2

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

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

Step 3.4.2.3

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

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

Step 3.4.2.4

Add $1$ and $1$ .

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

Step 3.4.2.5

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

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

Step 3.4.2.6

Rewrite $- 1 n$ as $- n$ .

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

Step 3.4.3

Reorder the factors of $(n^{2} - n) (n - 2) x^{n - 3}$ .

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

Step 4

Find the fourth derivative.

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Step 4.1

Since $(n - 2) (n^{2} - n)$ is constant with respect to $x$ , the derivative of $x^{n - 3} (n - 2) (n^{2} - n)$ with respect to $x$ is $(n - 2) (n^{2} - n) \frac{d}{d x} [x^{n - 3}]$ .

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

Step 4.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = n - 3$ .

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

Step 4.3

Simplify the expression.

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Step 4.3.1

Subtract $1$ from $- 3$ .

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

Step 4.3.2

Reorder the factors of $(n - 2) (n^{2} - n) ((n - 3) x^{n - 4})$ .

$f^{4} (x) = x^{n - 4} (n - 2) (n - 3) (n^{2} - n)$