Calculus Examples

$f (x) = 2 {(6)}^{3} - 24 {(6)}^{2} + 90 x + 8$

Step 1

Differentiate using the Sum Rule.

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Step 1.1

Raise $6$ to the power of $3$ .

$\frac{d}{d x} [2 \cdot 216 - 24 {(6)}^{2} + 90 x + 8]$

Step 1.2

Raise $6$ to the power of $2$ .

$\frac{d}{d x} [2 \cdot 216 - 24 \cdot 36 + 90 x + 8]$

Step 1.3

By the Sum Rule, the derivative of $2 \cdot 216 - 24 \cdot 36 + 90 x + 8$ with respect to $x$ is $\frac{d}{d x} [2 \cdot 216] + \frac{d}{d x} [- 24 \cdot 36] + \frac{d}{d x} [90 x] + \frac{d}{d x} [8]$ .

$\frac{d}{d x} [2 \cdot 216] + \frac{d}{d x} [- 24 \cdot 36] + \frac{d}{d x} [90 x] + \frac{d}{d x} [8]$

Step 2

Evaluate $\frac{d}{d x} [2 \cdot 216]$ .

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Step 2.1

Multiply $2$ by $216$ .

$\frac{d}{d x} [432] + \frac{d}{d x} [- 24 \cdot 36] + \frac{d}{d x} [90 x] + \frac{d}{d x} [8]$

Step 2.2

Since $432$ is constant with respect to $x$ , the derivative of $432$ with respect to $x$ is $0$ .

$0 + \frac{d}{d x} [- 24 \cdot 36] + \frac{d}{d x} [90 x] + \frac{d}{d x} [8]$

Step 3

Evaluate $\frac{d}{d x} [- 24 \cdot 36]$ .

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Step 3.1

Multiply $- 24$ by $36$ .

$0 + \frac{d}{d x} [- 864] + \frac{d}{d x} [90 x] + \frac{d}{d x} [8]$

Step 3.2

Since $- 864$ is constant with respect to $x$ , the derivative of $- 864$ with respect to $x$ is $0$ .

$0 + 0 + \frac{d}{d x} [90 x] + \frac{d}{d x} [8]$

Step 4

Evaluate $\frac{d}{d x} [90 x]$ .

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Step 4.1

Since $90$ is constant with respect to $x$ , the derivative of $90 x$ with respect to $x$ is $90 \frac{d}{d x} [x]$ .

$0 + 0 + 90 \frac{d}{d x} [x] + \frac{d}{d x} [8]$

Step 4.2

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

$0 + 0 + 90 \cdot 1 + \frac{d}{d x} [8]$

Step 4.3

Multiply $90$ by $1$ .

$0 + 0 + 90 + \frac{d}{d x} [8]$

Step 5

Since $8$ is constant with respect to $x$ , the derivative of $8$ with respect to $x$ is $0$ .

$0 + 0 + 90 + 0$

Step 6

Combine terms.

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Step 6.1

Add $0$ and $0$ .

$0 + 90 + 0$

Step 6.2

Add $0$ and $90$ .

$90 + 0$

Step 6.3

Add $90$ and $0$ .

$90$