Calculus Examples

$y = 6 x - 12 {(4 x^{2} - 3)}^{2}$ ; $x = 1$

Step 1

By the Sum Rule, the derivative of $6 x - 12 {(4 x^{2} - 3)}^{2}$ with respect to $x$ is $\frac{d}{d x} [6 x] + \frac{d}{d x} [- 12 {(4 x^{2} - 3)}^{2}]$ .

$\frac{d}{d x} [6 x] + \frac{d}{d x} [- 12 {(4 x^{2} - 3)}^{2}]$

Step 2

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

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Step 2.1

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

$6 \frac{d}{d x} [x] + \frac{d}{d x} [- 12 {(4 x^{2} - 3)}^{2}]$

Step 2.2

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

$6 \cdot 1 + \frac{d}{d x} [- 12 {(4 x^{2} - 3)}^{2}]$

Step 2.3

Multiply $6$ by $1$ .

$6 + \frac{d}{d x} [- 12 {(4 x^{2} - 3)}^{2}]$

Step 3

Evaluate $\frac{d}{d x} [- 12 {(4 x^{2} - 3)}^{2}]$ .

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Step 3.1

Since $- 12$ is constant with respect to $x$ , the derivative of $- 12 {(4 x^{2} - 3)}^{2}$ with respect to $x$ is $- 12 \frac{d}{d x} [{(4 x^{2} - 3)}^{2}]$ .

$6 - 12 \frac{d}{d x} [{(4 x^{2} - 3)}^{2}]$

Step 3.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{2}$ and $g (x) = 4 x^{2} - 3$ .

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Step 3.2.1

To apply the Chain Rule, set $u$ as $4 x^{2} - 3$ .

$6 - 12 (\frac{d}{d u} [u^{2}] \frac{d}{d x} [4 x^{2} - 3])$

Step 3.2.2

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

$6 - 12 (2 u \frac{d}{d x} [4 x^{2} - 3])$

Step 3.2.3

Replace all occurrences of $u$ with $4 x^{2} - 3$ .

$6 - 12 (2 (4 x^{2} - 3) \frac{d}{d x} [4 x^{2} - 3])$

Step 3.3

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

$6 - 12 (2 (4 x^{2} - 3) (\frac{d}{d x} [4 x^{2}] + \frac{d}{d x} [- 3]))$

Step 3.4

Since $4$ is constant with respect to $x$ , the derivative of $4 x^{2}$ with respect to $x$ is $4 \frac{d}{d x} [x^{2}]$ .

$6 - 12 (2 (4 x^{2} - 3) (4 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 3]))$

Step 3.5

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

$6 - 12 (2 (4 x^{2} - 3) (4 (2 x) + \frac{d}{d x} [- 3]))$

Step 3.6

Since $- 3$ is constant with respect to $x$ , the derivative of $- 3$ with respect to $x$ is $0$ .

$6 - 12 (2 (4 x^{2} - 3) (4 (2 x) + 0))$

Step 3.7

Multiply $2$ by $4$ .

$6 - 12 (2 (4 x^{2} - 3) (8 x + 0))$

Step 3.8

Add $8 x$ and $0$ .

$6 - 12 (2 (4 x^{2} - 3) (8 x))$

Step 3.9

Multiply $8$ by $2$ .

$6 - 12 (16 (4 x^{2} - 3) x)$

Step 3.10

Multiply $16$ by $- 12$ .

$6 - 192 (4 x^{2} - 3) x$

Step 4

Simplify.

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Step 4.1

Apply the distributive property.

$6 + (- 192 (4 x^{2}) - 192 \cdot - 3) x$

Step 4.2

Apply the distributive property.

$6 - 192 (4 x^{2}) x - 192 \cdot - 3 x$

Step 4.3

Combine terms.

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Step 4.3.1

Multiply $4$ by $- 192$ .

$6 - 768 x^{2} x - 192 \cdot - 3 x$

Step 4.3.2

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

$6 - 768 (x^{1} x^{2}) - 192 \cdot - 3 x$

Step 4.3.3

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

$6 - 768 x^{1 + 2} - 192 \cdot - 3 x$

Step 4.3.4

Add $1$ and $2$ .

$6 - 768 x^{3} - 192 \cdot - 3 x$

Step 4.3.5

Multiply $- 192$ by $- 3$ .

$6 - 768 x^{3} + 576 x$

Step 4.4

Reorder terms.

$- 768 x^{3} + 576 x + 6$

Step 5

Evaluate the derivative at $x = 1$ .

$- 768 {(1)}^{3} + 576 (1) + 6$

Step 6

Simplify.

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Step 6.1

Simplify each term.

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Step 6.1.1

One to any power is one.

$- 768 \cdot 1 + 576 (1) + 6$

Step 6.1.2

Multiply $- 768$ by $1$ .

$- 768 + 576 (1) + 6$

Step 6.1.3

Multiply $576$ by $1$ .

$- 768 + 576 + 6$

Step 6.2

Simplify by adding numbers.

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Step 6.2.1

Add $- 768$ and $576$ .

$- 192 + 6$

Step 6.2.2

Add $- 192$ and $6$ .

$- 186$