Calculus Examples

$y = 9 x^{4} + a x^{3} + b x^{2} + c x + d$

Step 1

Find the first derivative.

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Step 1.1

By the Sum Rule, the derivative of $9 x^{4} + a x^{3} + b x^{2} + c x + d$ with respect to $x$ is $\frac{d}{d x} [9 x^{4}] + \frac{d}{d x} [a x^{3}] + \frac{d}{d x} [b x^{2}] + \frac{d}{d x} [c x] + \frac{d}{d x} [d]$ .

$\frac{d}{d x} [9 x^{4}] + \frac{d}{d x} [a x^{3}] + \frac{d}{d x} [b x^{2}] + \frac{d}{d x} [c x] + \frac{d}{d x} [d]$

Step 1.2

Evaluate $\frac{d}{d x} [9 x^{4}]$ .

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Step 1.2.1

Since $9$ is constant with respect to $x$ , the derivative of $9 x^{4}$ with respect to $x$ is $9 \frac{d}{d x} [x^{4}]$ .

$9 \frac{d}{d x} [x^{4}] + \frac{d}{d x} [a x^{3}] + \frac{d}{d x} [b x^{2}] + \frac{d}{d x} [c x] + \frac{d}{d x} [d]$

Step 1.2.2

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

$9 (4 x^{3}) + \frac{d}{d x} [a x^{3}] + \frac{d}{d x} [b x^{2}] + \frac{d}{d x} [c x] + \frac{d}{d x} [d]$

Step 1.2.3

Multiply $4$ by $9$ .

$36 x^{3} + \frac{d}{d x} [a x^{3}] + \frac{d}{d x} [b x^{2}] + \frac{d}{d x} [c x] + \frac{d}{d x} [d]$

Step 1.3

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

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Step 1.3.1

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

$36 x^{3} + a \frac{d}{d x} [x^{3}] + \frac{d}{d x} [b x^{2}] + \frac{d}{d x} [c x] + \frac{d}{d x} [d]$

Step 1.3.2

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

$36 x^{3} + a (3 x^{2}) + \frac{d}{d x} [b x^{2}] + \frac{d}{d x} [c x] + \frac{d}{d x} [d]$

Step 1.3.3

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

$36 x^{3} + 3 a x^{2} + \frac{d}{d x} [b x^{2}] + \frac{d}{d x} [c x] + \frac{d}{d x} [d]$

Step 1.4

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

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Step 1.4.1

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

$36 x^{3} + 3 a x^{2} + b \frac{d}{d x} [x^{2}] + \frac{d}{d x} [c x] + \frac{d}{d x} [d]$

Step 1.4.2

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

$36 x^{3} + 3 a x^{2} + b (2 x) + \frac{d}{d x} [c x] + \frac{d}{d x} [d]$

Step 1.4.3

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

$36 x^{3} + 3 a x^{2} + 2 b x + \frac{d}{d x} [c x] + \frac{d}{d x} [d]$

Step 1.5

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

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Step 1.5.1

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

$36 x^{3} + 3 a x^{2} + 2 b x + c \frac{d}{d x} [x] + \frac{d}{d x} [d]$

Step 1.5.2

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

$36 x^{3} + 3 a x^{2} + 2 b x + c \cdot 1 + \frac{d}{d x} [d]$

Step 1.5.3

Multiply $c$ by $1$ .

$36 x^{3} + 3 a x^{2} + 2 b x + c + \frac{d}{d x} [d]$

Step 1.6

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

$36 x^{3} + 3 a x^{2} + 2 b x + c + 0$

Step 1.7

Simplify.

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Step 1.7.1

Add $36 x^{3} + 3 a x^{2} + 2 b x + c$ and $0$ .

$36 x^{3} + 3 a x^{2} + 2 b x + c$

Step 1.7.2

Reorder terms.

$f' (x) = 36 x^{3} + c + 3 x^{2} a + 2 b x$

Step 2

Find the second derivative.

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Step 2.1

By the Sum Rule, the derivative of $36 x^{3} + c + 3 x^{2} a + 2 b x$ with respect to $x$ is $\frac{d}{d x} [36 x^{3}] + \frac{d}{d x} [c] + \frac{d}{d x} [3 x^{2} a] + \frac{d}{d x} [2 b x]$ .

$\frac{d}{d x} [36 x^{3}] + \frac{d}{d x} [c] + \frac{d}{d x} [3 x^{2} a] + \frac{d}{d x} [2 b x]$

Step 2.2

Evaluate $\frac{d}{d x} [36 x^{3}]$ .

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Step 2.2.1

Since $36$ is constant with respect to $x$ , the derivative of $36 x^{3}$ with respect to $x$ is $36 \frac{d}{d x} [x^{3}]$ .

$36 \frac{d}{d x} [x^{3}] + \frac{d}{d x} [c] + \frac{d}{d x} [3 x^{2} a] + \frac{d}{d x} [2 b x]$

Step 2.2.2

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

$36 (3 x^{2}) + \frac{d}{d x} [c] + \frac{d}{d x} [3 x^{2} a] + \frac{d}{d x} [2 b x]$

Step 2.2.3

Multiply $3$ by $36$ .

$108 x^{2} + \frac{d}{d x} [c] + \frac{d}{d x} [3 x^{2} a] + \frac{d}{d x} [2 b x]$

Step 2.3

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

$108 x^{2} + 0 + \frac{d}{d x} [3 x^{2} a] + \frac{d}{d x} [2 b x]$

Step 2.4

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

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Step 2.4.1

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

$108 x^{2} + 0 + 3 a \frac{d}{d x} [x^{2}] + \frac{d}{d x} [2 b x]$

Step 2.4.2

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

$108 x^{2} + 0 + 3 a (2 x) + \frac{d}{d x} [2 b x]$

Step 2.4.3

Multiply $2$ by $3$ .

$108 x^{2} + 0 + 6 a x + \frac{d}{d x} [2 b x]$

Step 2.5

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

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Step 2.5.1

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

$108 x^{2} + 0 + 6 a x + 2 b \frac{d}{d x} [x]$

Step 2.5.2

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

$108 x^{2} + 0 + 6 a x + 2 b \cdot 1$

Step 2.5.3

Multiply $2$ by $1$ .

$108 x^{2} + 0 + 6 a x + 2 b$

Step 2.6

Simplify.

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Step 2.6.1

Add $108 x^{2}$ and $0$ .

$108 x^{2} + 6 a x + 2 b$

Step 2.6.2

Reorder terms.

$f'' (x) = 108 x^{2} + 2 b + 6 a x$

Step 3

Find the third derivative.

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Step 3.1

By the Sum Rule, the derivative of $108 x^{2} + 2 b + 6 a x$ with respect to $x$ is $\frac{d}{d x} [108 x^{2}] + \frac{d}{d x} [2 b] + \frac{d}{d x} [6 a x]$ .

$\frac{d}{d x} [108 x^{2}] + \frac{d}{d x} [2 b] + \frac{d}{d x} [6 a x]$

Step 3.2

Evaluate $\frac{d}{d x} [108 x^{2}]$ .

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Step 3.2.1

Since $108$ is constant with respect to $x$ , the derivative of $108 x^{2}$ with respect to $x$ is $108 \frac{d}{d x} [x^{2}]$ .

$108 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [2 b] + \frac{d}{d x} [6 a x]$

Step 3.2.2

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

$108 (2 x) + \frac{d}{d x} [2 b] + \frac{d}{d x} [6 a x]$

Step 3.2.3

Multiply $2$ by $108$ .

$216 x + \frac{d}{d x} [2 b] + \frac{d}{d x} [6 a x]$

Step 3.3

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

$216 x + 0 + \frac{d}{d x} [6 a x]$

Step 3.4

Evaluate $\frac{d}{d x} [6 a x]$ .

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Step 3.4.1

Since $6 a$ is constant with respect to $x$ , the derivative of $6 a x$ with respect to $x$ is $6 a \frac{d}{d x} [x]$ .

$216 x + 0 + 6 a \frac{d}{d x} [x]$

Step 3.4.2

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

$216 x + 0 + 6 a \cdot 1$

Step 3.4.3

Multiply $6$ by $1$ .

$216 x + 0 + 6 a$

Step 3.5

Simplify.

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Step 3.5.1

Add $216 x$ and $0$ .

$216 x + 6 a$

Step 3.5.2

Reorder terms.

$f''' (x) = 6 a + 216 x$