Calculus Examples

$y = x^{4} + 2 x^{2} - x$

Step 1

Set $y$ as a function of $x$ .

$f (x) = x^{4} + 2 x^{2} - x$

Step 2

Find the derivative.

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Step 2.1

Differentiate.

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Step 2.1.1

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

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

Step 2.1.2

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

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

Step 2.2

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

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Step 2.2.1

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

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

Step 2.2.2

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

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

Step 2.2.3

Multiply $2$ by $2$ .

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

Step 2.3

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

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Step 2.3.1

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

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

Step 2.3.2

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

$4 x^{3} + 4 x - 1 \cdot 1$

Step 2.3.3

Multiply $- 1$ by $1$ .

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

Step 3

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

$x \approx 0.2367329$

Step 4

Solve the original function $f (x) = x^{4} + 2 x^{2} - x$ at $x = 0.2367329$ .

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Step 4.1

Replace the variable $x$ with $0.2367329$ in the expression.

$f (0.2367329) = {(0.2367329)}^{4} + 2 {(0.2367329)}^{2} - (0.2367329)$

Step 4.2

Simplify the result.

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Step 4.2.1

Simplify each term.

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Step 4.2.1.1

Raise $0.2367329$ to the power of $4$ .

$f (0.2367329) = 0.00314075 + 2 {(0.2367329)}^{2} - (0.2367329)$

Step 4.2.1.2

Raise $0.2367329$ to the power of $2$ .

$f (0.2367329) = 0.00314075 + 2 \cdot 0.05604246 - (0.2367329)$

Step 4.2.1.3

Multiply $2$ by $0.05604246$ .

$f (0.2367329) = 0.00314075 + 0.11208493 - (0.2367329)$

Step 4.2.1.4

Multiply $- 1$ by $0.2367329$ .

$f (0.2367329) = 0.00314075 + 0.11208493 - 0.2367329$

Step 4.2.2

Simplify by adding and subtracting.

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Step 4.2.2.1

Add $0.00314075$ and $0.11208493$ .

$f (0.2367329) = 0.11522569 - 0.2367329$

Step 4.2.2.2

Subtract $0.2367329$ from $0.11522569$ .

$f (0.2367329) = - 0.12150721$

Step 4.2.3

The final answer is $- 0.12150721$ .

$- 0.12150721$

Step 5

The horizontal tangent line on function $f (x) = x^{4} + 2 x^{2} - x$ is $y = - 0.12150721$ .

$y = - 0.12150721$

Step 6