Calculus Examples

$f (x) = a x^{4} + b x^{3} + c x^{2} + d x + e$

Step 1

Find the first derivative of the function.

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Step 1.1

By the Sum Rule, the derivative of $a x^{4} + b x^{3} + c x^{2} + d x + e$ with respect to $x$ is $\frac{d}{d x} [a x^{4}] + \frac{d}{d x} [b x^{3}] + \frac{d}{d x} [c x^{2}] + \frac{d}{d x} [d x] + \frac{d}{d x} [e]$ .

$\frac{d}{d x} [a x^{4}] + \frac{d}{d x} [b x^{3}] + \frac{d}{d x} [c x^{2}] + \frac{d}{d x} [d x] + \frac{d}{d x} [e]$

Step 1.2

Evaluate $\frac{d}{d x} [a x^{4}]$ .

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Step 1.2.1

Since $a$ is constant with respect to $x$ , the derivative of $a x^{4}$ with respect to $x$ is $a \frac{d}{d x} [x^{4}]$ .

$a \frac{d}{d x} [x^{4}] + \frac{d}{d x} [b x^{3}] + \frac{d}{d x} [c x^{2}] + \frac{d}{d x} [d x] + \frac{d}{d x} [e]$

Step 1.2.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 4$ .

$a (4 x^{3}) + \frac{d}{d x} [b x^{3}] + \frac{d}{d x} [c x^{2}] + \frac{d}{d x} [d x] + \frac{d}{d x} [e]$

Step 1.2.3

Move $4$ to the left of $a$ .

$4 a x^{3} + \frac{d}{d x} [b x^{3}] + \frac{d}{d x} [c x^{2}] + \frac{d}{d x} [d x] + \frac{d}{d x} [e]$

Step 1.3

Evaluate $\frac{d}{d x} [b x^{3}]$ .

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Step 1.3.1

Since $b$ is constant with respect to $x$ , the derivative of $b x^{3}$ with respect to $x$ is $b \frac{d}{d x} [x^{3}]$ .

$4 a x^{3} + b \frac{d}{d x} [x^{3}] + \frac{d}{d x} [c x^{2}] + \frac{d}{d x} [d x] + \frac{d}{d x} [e]$

Step 1.3.2

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

$4 a x^{3} + b (3 x^{2}) + \frac{d}{d x} [c x^{2}] + \frac{d}{d x} [d x] + \frac{d}{d x} [e]$

Step 1.3.3

Move $3$ to the left of $b$ .

$4 a x^{3} + 3 b x^{2} + \frac{d}{d x} [c x^{2}] + \frac{d}{d x} [d x] + \frac{d}{d x} [e]$

Step 1.4

Evaluate $\frac{d}{d x} [c x^{2}]$ .

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Step 1.4.1

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

$4 a x^{3} + 3 b x^{2} + c \frac{d}{d x} [x^{2}] + \frac{d}{d x} [d x] + \frac{d}{d x} [e]$

Step 1.4.2

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

$4 a x^{3} + 3 b x^{2} + c (2 x) + \frac{d}{d x} [d x] + \frac{d}{d x} [e]$

Step 1.4.3

Move $2$ to the left of $c$ .

$4 a x^{3} + 3 b x^{2} + 2 c x + \frac{d}{d x} [d x] + \frac{d}{d x} [e]$

Step 1.5

Evaluate $\frac{d}{d x} [d x]$ .

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Step 1.5.1

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

$4 a x^{3} + 3 b x^{2} + 2 c x + d \frac{d}{d x} [x] + \frac{d}{d x} [e]$

Step 1.5.2

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

$4 a x^{3} + 3 b x^{2} + 2 c x + d \cdot 1 + \frac{d}{d x} [e]$

Step 1.5.3

Multiply $d$ by $1$ .

$4 a x^{3} + 3 b x^{2} + 2 c x + d + \frac{d}{d x} [e]$

Step 1.6

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

$4 a x^{3} + 3 b x^{2} + 2 c x + d + 0$

Step 1.7

Simplify.

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Step 1.7.1

Add $4 a x^{3} + 3 b x^{2} + 2 c x + d$ and $0$ .

$4 a x^{3} + 3 b x^{2} + 2 c x + d$

Step 1.7.2

Reorder terms.

$d + 4 x^{3} a + 3 x^{2} b + 2 c x$

Step 2

Find the second derivative of the function.

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Step 2.1

Differentiate.

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Step 2.1.1

By the Sum Rule, the derivative of $d + 4 x^{3} a + 3 x^{2} b + 2 c x$ with respect to $x$ is $\frac{d}{d x} [d] + \frac{d}{d x} [4 x^{3} a] + \frac{d}{d x} [3 x^{2} b] + \frac{d}{d x} [2 c x]$ .

$f'' (x) = \frac{d}{d x} (d) + \frac{d}{d x} (4 x^{3} a) + \frac{d}{d x} (3 x^{2} b) + \frac{d}{d x} (2 c x)$

Step 2.1.2

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

$f'' (x) = 0 + \frac{d}{d x} (4 x^{3} a) + \frac{d}{d x} (3 x^{2} b) + \frac{d}{d x} (2 c x)$

Step 2.2

Evaluate $\frac{d}{d x} [4 x^{3} a]$ .

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Step 2.2.1

Since $4 a$ is constant with respect to $x$ , the derivative of $4 x^{3} a$ with respect to $x$ is $4 a \frac{d}{d x} [x^{3}]$ .

$f'' (x) = 0 + 4 a \frac{d}{d x} (x^{3}) + \frac{d}{d x} (3 x^{2} b) + \frac{d}{d x} (2 c x)$

Step 2.2.2

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

$f'' (x) = 0 + 4 a (3 x^{2}) + \frac{d}{d x} (3 x^{2} b) + \frac{d}{d x} (2 c x)$

Step 2.2.3

Multiply $3$ by $4$ .

$f'' (x) = 0 + 12 a x^{2} + \frac{d}{d x} (3 x^{2} b) + \frac{d}{d x} (2 c x)$

Step 2.3

Evaluate $\frac{d}{d x} [3 x^{2} b]$ .

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Step 2.3.1

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

$f'' (x) = 0 + 12 a x^{2} + 3 b \frac{d}{d x} (x^{2}) + \frac{d}{d x} (2 c x)$

Step 2.3.2

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

$f'' (x) = 0 + 12 a x^{2} + 3 b (2 x) + \frac{d}{d x} (2 c x)$

Step 2.3.3

Multiply $2$ by $3$ .

$f'' (x) = 0 + 12 a x^{2} + 6 b x + \frac{d}{d x} (2 c x)$

Step 2.4

Evaluate $\frac{d}{d x} [2 c x]$ .

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Step 2.4.1

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

$f'' (x) = 0 + 12 a x^{2} + 6 b x + 2 c \frac{d}{d x} (x)$

Step 2.4.2

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

$f'' (x) = 0 + 12 a x^{2} + 6 b x + 2 c \cdot 1$

Step 2.4.3

Multiply $2$ by $1$ .

$f'' (x) = 0 + 12 a x^{2} + 6 b x + 2 c$

Step 2.5

Simplify.

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Step 2.5.1

Add $0$ and $12 a x^{2}$ .

$f'' (x) = 12 a x^{2} + 6 b x + 2 c$

Step 2.5.2

Reorder terms.

$f'' (x) = 2 c + 12 x^{2} a + 6 b x$

Step 3

To find the local maximum and minimum values of the function, set the derivative equal to $0$ and solve.

$d + 4 x^{3} a + 3 x^{2} b + 2 c x = 0$

Step 4

Find the first derivative.

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Step 4.1

Find the first derivative.

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Step 4.1.1

By the Sum Rule, the derivative of $a x^{4} + b x^{3} + c x^{2} + d x + e$ with respect to $a$ is $\frac{d}{d a} [a x^{4}] + \frac{d}{d a} [b x^{3}] + \frac{d}{d a} [c x^{2}] + \frac{d}{d a} [d x] + \frac{d}{d a} [e]$ .

$f' (a) = \frac{d}{d a} (a x^{4}) + \frac{d}{d a} (b x^{3}) + \frac{d}{d a} (c x^{2}) + \frac{d}{d a} (d x) + \frac{d}{d a} (e)$

Step 4.1.2

Evaluate $\frac{d}{d a} [a x^{4}]$ .

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Step 4.1.2.1

Since $x^{4}$ is constant with respect to $a$ , the derivative of $a x^{4}$ with respect to $a$ is $x^{4} \frac{d}{d a} [a]$ .

$f' (a) = x^{4} \frac{d}{d a} (a) + \frac{d}{d a} (b x^{3}) + \frac{d}{d a} (c x^{2}) + \frac{d}{d a} (d x) + \frac{d}{d a} (e)$

Step 4.1.2.2

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

$f' (a) = x^{4} \cdot 1 + \frac{d}{d a} (b x^{3}) + \frac{d}{d a} (c x^{2}) + \frac{d}{d a} (d x) + \frac{d}{d a} (e)$

Step 4.1.2.3

Multiply $x^{4}$ by $1$ .

$f' (a) = x^{4} + \frac{d}{d a} (b x^{3}) + \frac{d}{d a} (c x^{2}) + \frac{d}{d a} (d x) + \frac{d}{d a} (e)$

Step 4.1.3

Differentiate using the Constant Rule.

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Step 4.1.3.1

Since $b x^{3}$ is constant with respect to $a$ , the derivative of $b x^{3}$ with respect to $a$ is $0$ .

$f' (a) = x^{4} + 0 + \frac{d}{d a} (c x^{2}) + \frac{d}{d a} (d x) + \frac{d}{d a} (e)$

Step 4.1.3.2

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

$f' (a) = x^{4} + 0 + 0 + \frac{d}{d a} (d x) + \frac{d}{d a} (e)$

Step 4.1.3.3

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

$f' (a) = x^{4} + 0 + 0 + 0 + \frac{d}{d a} (e)$

Step 4.1.3.4

Since $e$ is constant with respect to $a$ , the derivative of $e$ with respect to $a$ is $0$ .

$f' (a) = x^{4} + 0 + 0 + 0 + 0$

Step 4.1.4

Combine terms.

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Step 4.1.4.1

Add $x^{4}$ and $0$ .

$f' (a) = x^{4} + 0 + 0 + 0$

Step 4.1.4.2

Add $x^{4}$ and $0$ .

$f' (a) = x^{4} + 0 + 0$

Step 4.1.4.3

Add $x^{4}$ and $0$ .

$f' (a) = x^{4} + 0$

Step 4.1.4.4

Add $x^{4}$ and $0$ .

$f' (a) = x^{4}$

Step 4.2

The first derivative of $f (x)$ with respect to $x$ is $x^{4}$ .

$x^{4}$

Step 5

Set the first derivative equal to $0$ then solve the equation $x^{4} = 0$ .

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Step 5.1

Set the first derivative equal to $0$ .

$x^{4} = 0$

Step 5.2

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt[4]{0}$

Step 5.3

Simplify $\pm \sqrt[4]{0}$ .

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Step 5.3.1

Rewrite $0$ as $0^{4}$ .

$x = \pm \sqrt[4]{0^{4}}$

Step 5.3.2

Pull terms out from under the radical, assuming positive real numbers.

$x = \pm 0$

Step 5.3.3

Plus or minus $0$ is $0$ .

$x = 0$

Step 6

Find the values where the derivative is undefined.

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Step 6.1

The domain of the expression is all real numbers except where the expression is undefined. In this case, there is no real number that makes the expression undefined.

Step 7

Critical points to evaluate.

$x = 0$

Step 8

Evaluate the second derivative at $x = 0$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$2 c + 12 {(0)}^{2} a + 6 b (0)$

Step 9

Evaluate the second derivative.

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Step 9.1

Simplify each term.

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Step 9.1.1

Raising $0$ to any positive power yields $0$ .

$2 c + 12 \cdot 0 a + 6 b (0)$

Step 9.1.2

Multiply $12$ by $0$ .

$2 c + 0 a + 6 b (0)$

Step 9.1.3

Multiply $0$ by $a$ .

$2 c + 0 + 6 b (0)$

Step 9.1.4

Multiply $6 b (0)$ .

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Step 9.1.4.1

Multiply $0$ by $6$ .

$2 c + 0 + 0 b$

Step 9.1.4.2

Multiply $0$ by $b$ .

$2 c + 0 + 0$

Step 9.2

Combine the opposite terms in $2 c + 0 + 0$ .

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Step 9.2.1

Add $2 c$ and $0$ .

$2 c + 0$

Step 9.2.2

Add $2 c$ and $0$ .

$2 c$

Step 10

Since the first derivative test failed, there are no local extrema.

No Local Extrema

Step 11