Calculus Examples

$f (x) = 4 x^{3} - x^{4} + x + 5$

Find the first derivative.

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By the Sum Rule, the derivative of $4 x^{3} - x^{4} + x + 5$ with respect to $x$ is $\frac{d}{d x} [4 x^{3}] + \frac{d}{d x} [- x^{4}] + \frac{d}{d x} [x] + \frac{d}{d x} [5]$ .

$\frac{d}{d x} [4 x^{3}] + \frac{d}{d x} [- x^{4}] + \frac{d}{d x} [x] + \frac{d}{d x} [5]$

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

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

$4 \frac{d}{d x} [x^{3}] + \frac{d}{d x} [- x^{4}] + \frac{d}{d x} [x] + \frac{d}{d x} [5]$

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

$4 (3 x^{2}) + \frac{d}{d x} [- x^{4}] + \frac{d}{d x} [x] + \frac{d}{d x} [5]$

Multiply $3$ by $4$ .

$12 x^{2} + \frac{d}{d x} [- x^{4}] + \frac{d}{d x} [x] + \frac{d}{d x} [5]$

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

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Since $- 1$ is constant with respect to $x$ , the derivative of $- x^{4}$ with respect to $x$ is $- \frac{d}{d x} [x^{4}]$ .

$12 x^{2} - \frac{d}{d x} [x^{4}] + \frac{d}{d x} [x] + \frac{d}{d x} [5]$

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

$12 x^{2} - (4 x^{3}) + \frac{d}{d x} [x] + \frac{d}{d x} [5]$

Multiply $4$ by $- 1$ .

$12 x^{2} - 4 x^{3} + \frac{d}{d x} [x] + \frac{d}{d x} [5]$

Differentiate.

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

$12 x^{2} - 4 x^{3} + 1 + \frac{d}{d x} [5]$

Since $5$ is constant with respect to $x$ , the derivative of $5$ with respect to $x$ is $0$ .

$12 x^{2} - 4 x^{3} + 1 + 0$

Simplify.

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Add $12 x^{2} - 4 x^{3} + 1$ and $0$ .

$12 x^{2} - 4 x^{3} + 1$

Reorder terms.

$- 4 x^{3} + 12 x^{2} + 1$

Graph each side of the equation. The solution is the x-value of the point of intersection.

$x \approx 3.02727941$

Split $(- \infty, \infty)$ into separate intervals around the $x$ values that make the first derivative $0$ or undefined.

$(- \infty, 3.02727941) \cup (3.02727941, \infty)$

Substitute any number, such as $0$ , from the interval $(- \infty, 3.02727941)$ in the first derivative $- 4 x^{3} + 12 x^{2} + 1$ to check if the result is negative or positive.

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Replace the variable $x$ with $0$ in the expression.

$f' (0) = - 4 {(0)}^{3} + 12 {(0)}^{2} + 1$

Simplify the result.

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Simplify each term.

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Raising $0$ to any positive power yields $0$ .

$f' (0) = - 4 \cdot 0 + 12 {(0)}^{2} + 1$

Multiply $- 4$ by $0$ .

$f' (0) = 0 + 12 {(0)}^{2} + 1$

Raising $0$ to any positive power yields $0$ .

$f' (0) = 0 + 12 \cdot 0 + 1$

Multiply $12$ by $0$ .

$f' (0) = 0 + 0 + 1$

Simplify by adding numbers.

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Add $0$ and $0$ .

$f' (0) = 0 + 1$

Add $0$ and $1$ .

$f' (0) = 1$

The final answer is $1$ .

$1$

Substitute any number, such as $6$ , from the interval $(3.02727941, \infty)$ in the first derivative $- 4 x^{3} + 12 x^{2} + 1$ to check if the result is negative or positive.

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Replace the variable $x$ with $6$ in the expression.

$f' (6) = - 4 {(6)}^{3} + 12 {(6)}^{2} + 1$

Simplify the result.

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Simplify each term.

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

$f' (6) = - 4 \cdot 216 + 12 {(6)}^{2} + 1$

Multiply $- 4$ by $216$ .

$f' (6) = - 864 + 12 {(6)}^{2} + 1$

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

$f' (6) = - 864 + 12 \cdot 36 + 1$

Multiply $12$ by $36$ .

$f' (6) = - 864 + 432 + 1$

Simplify by adding numbers.

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Add $- 864$ and $432$ .

$f' (6) = - 432 + 1$

Add $- 432$ and $1$ .

$f' (6) = - 431$

The final answer is $- 431$ .

$- 431$

Since the first derivative changed signs from positive to negative around $x = 3.02727941$ , then there is a turning point at $x = 3.02727941$ .

Find the y-coordinate of $3.02727941$ to find the turning point.

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Find $f (3.02727941)$ to find the y-coordinate of $3.02727941$ .

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Replace the variable $x$ with $3.02727941$ in the expression.

$f (3.02727941) = 4 {(3.02727941)}^{3} - {(3.02727941)}^{4} + 3.02727941 + 5$

Simplify $4 {(3.02727941)}^{3} - {(3.02727941)}^{4} + 3.02727941 + 5$ .

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Remove parentheses.

$4 {(3.02727941)}^{3} - {(3.02727941)}^{4} + 3.02727941 + 5$

Simplify each term.

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Raise $3.02727941$ to the power of $3$ .

$4 \cdot 27.7432619 - {(3.02727941)}^{4} + 3.02727941 + 5$

Multiply $4$ by $27.7432619$ .

$110.9730476 - {(3.02727941)}^{4} + 3.02727941 + 5$

Raise $3.02727941$ to the power of $4$ .

$110.9730476 - 1 \cdot 83.98660555 + 3.02727941 + 5$

Multiply $- 1$ by $83.98660555$ .

$110.9730476 - 83.98660555 + 3.02727941 + 5$

Simplify by adding and subtracting.

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Subtract $83.98660555$ from $110.9730476$ .

$26.98644204 + 3.02727941 + 5$

Add $26.98644204$ and $3.02727941$ .

$30.01372146 + 5$

Add $30.01372146$ and $5$ .

$35.01372146$

Write the $x$ and $y$ coordinates in point form.

$(3.02727941, 35.01372146)$

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