Calculus Examples

$f (x) = 4 x^{3} - 7 x^{2} - 4$ , $(4, 140)$

Step 1

Find the derivative of the function. To find the slope of the equation tangent to the line, evaluate the derivative at the desired value of $x$ .

$\frac{d}{d x} (4 x^{3} - 7 x^{2} - 4)$

Step 2

By the Sum Rule, the derivative of $4 x^{3} - 7 x^{2} - 4$ with respect to $x$ is $\frac{d}{d x} [4 x^{3}] + \frac{d}{d x} [- 7 x^{2}] + \frac{d}{d x} [- 4]$ .

$\frac{d}{d x} [4 x^{3}] + \frac{d}{d x} [- 7 x^{2}] + \frac{d}{d x} [- 4]$

Step 3

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

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Step 3.1

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

$4 \frac{d}{d x} [x^{3}] + \frac{d}{d x} [- 7 x^{2}] + \frac{d}{d x} [- 4]$

Step 3.2

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

$4 (3 x^{2}) + \frac{d}{d x} [- 7 x^{2}] + \frac{d}{d x} [- 4]$

Step 3.3

Multiply $3$ by $4$ .

$12 x^{2} + \frac{d}{d x} [- 7 x^{2}] + \frac{d}{d x} [- 4]$

Step 4

Evaluate $\frac{d}{d x} [- 7 x^{2}]$ .

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Step 4.1

Since $- 7$ is constant with respect to $x$ , the derivative of $- 7 x^{2}$ with respect to $x$ is $- 7 \frac{d}{d x} [x^{2}]$ .

$12 x^{2} - 7 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 4]$

Step 4.2

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

$12 x^{2} - 7 (2 x) + \frac{d}{d x} [- 4]$

Step 4.3

Multiply $2$ by $- 7$ .

$12 x^{2} - 14 x + \frac{d}{d x} [- 4]$

Step 5

Differentiate using the Constant Rule.

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Step 5.1

Since $- 4$ is constant with respect to $x$ , the derivative of $- 4$ with respect to $x$ is $0$ .

$12 x^{2} - 14 x + 0$

Step 5.2

Add $12 x^{2} - 14 x$ and $0$ .

$12 x^{2} - 14 x$

Step 6

The derivative of the equation in terms of $y$ can also be represented as $f' (x)$ .

$f' (x) = 12 x^{2} - 14 x$

Step 7

Replace the variable $x$ with $4$ in the expression.

$f' (4) = 12 {(4)}^{2} - 14 \cdot 4$

Step 8

Simplify each term.

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Step 8.1

Raise $4$ to the power of $2$ .

$f' (4) = 12 \cdot 16 - 14 \cdot 4$

Step 8.2

Multiply $12$ by $16$ .

$f' (4) = 192 - 14 \cdot 4$

Step 8.3

Multiply $- 14$ by $4$ .

$f' (4) = 192 - 56$

Step 9

Subtract $56$ from $192$ .

$f' (4) = 136$

Step 10

The derivative at $(4, 140)$ is $136$ .

$136$