Calculus Examples

$y = 3 x^{2} - 3 x + 1$ , $x = - 4$

Step 1

By the Sum Rule, the derivative of $3 x^{2} - 3 x + 1$ with respect to $x$ is $\frac{d}{d x} [3 x^{2}] + \frac{d}{d x} [- 3 x] + \frac{d}{d x} [1]$ .

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

Step 2

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

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

Multiply $2$ by $3$ .

$6 x + \frac{d}{d x} [- 3 x] + \frac{d}{d x} [1]$

Step 3

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

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Step 3.1

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

$6 x - 3 \frac{d}{d x} [x] + \frac{d}{d x} [1]$

Step 3.2

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

$6 x - 3 \cdot 1 + \frac{d}{d x} [1]$

Step 3.3

Multiply $- 3$ by $1$ .

$6 x - 3 + \frac{d}{d x} [1]$

Step 4

Differentiate using the Constant Rule.

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Step 4.1

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

$6 x - 3 + 0$

Step 4.2

Add $6 x - 3$ and $0$ .

$6 x - 3$

Step 5

Evaluate the derivative at $x = - 4$ .

$6 (- 4) - 3$

Step 6

Simplify.

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Step 6.1

Multiply $6$ by $- 4$ .

$- 24 - 3$

Step 6.2

Subtract $3$ from $- 24$ .

$- 27$

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