Calculus Examples

$y = x^{4} - \frac{3}{2} x^{3} + 5 x^{2} - 6 x - 2$

Step 1

Differentiate.

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Step 1.1

By the Sum Rule, the derivative of $x^{4} - \frac{3}{2} x^{3} + 5 x^{2} - 6 x - 2$ with respect to $x$ is $\frac{d}{d x} [x^{4}] + \frac{d}{d x} [- \frac{3}{2} x^{3}] + \frac{d}{d x} [5 x^{2}] + \frac{d}{d x} [- 6 x] + \frac{d}{d x} [- 2]$ .

$\frac{d}{d x} [x^{4}] + \frac{d}{d x} [- \frac{3}{2} x^{3}] + \frac{d}{d x} [5 x^{2}] + \frac{d}{d x} [- 6 x] + \frac{d}{d x} [- 2]$

Step 1.2

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

$4 x^{3} + \frac{d}{d x} [- \frac{3}{2} x^{3}] + \frac{d}{d x} [5 x^{2}] + \frac{d}{d x} [- 6 x] + \frac{d}{d x} [- 2]$

Step 2

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

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Step 2.1

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

$4 x^{3} - \frac{3}{2} \frac{d}{d x} [x^{3}] + \frac{d}{d x} [5 x^{2}] + \frac{d}{d x} [- 6 x] + \frac{d}{d x} [- 2]$

Step 2.2

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

$4 x^{3} - \frac{3}{2} (3 x^{2}) + \frac{d}{d x} [5 x^{2}] + \frac{d}{d x} [- 6 x] + \frac{d}{d x} [- 2]$

Step 2.3

Multiply $3$ by $- 1$ .

$4 x^{3} - 3 (\frac{3}{2}) x^{2} + \frac{d}{d x} [5 x^{2}] + \frac{d}{d x} [- 6 x] + \frac{d}{d x} [- 2]$

Step 2.4

Combine $- 3$ and $\frac{3}{2}$ .

$4 x^{3} + \frac{- 3 \cdot 3}{2} x^{2} + \frac{d}{d x} [5 x^{2}] + \frac{d}{d x} [- 6 x] + \frac{d}{d x} [- 2]$

Step 2.5

Multiply $- 3$ by $3$ .

$4 x^{3} + \frac{- 9}{2} x^{2} + \frac{d}{d x} [5 x^{2}] + \frac{d}{d x} [- 6 x] + \frac{d}{d x} [- 2]$

Step 2.6

Combine $\frac{- 9}{2}$ and $x^{2}$ .

$4 x^{3} + \frac{- 9 x^{2}}{2} + \frac{d}{d x} [5 x^{2}] + \frac{d}{d x} [- 6 x] + \frac{d}{d x} [- 2]$

Step 2.7

Move the negative in front of the fraction.

$4 x^{3} - \frac{9 x^{2}}{2} + \frac{d}{d x} [5 x^{2}] + \frac{d}{d x} [- 6 x] + \frac{d}{d x} [- 2]$

Step 3

Evaluate $\frac{d}{d x} [5 x^{2}]$ .

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Step 3.1

Since $5$ is constant with respect to $x$ , the derivative of $5 x^{2}$ with respect to $x$ is $5 \frac{d}{d x} [x^{2}]$ .

$4 x^{3} - \frac{9 x^{2}}{2} + 5 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 6 x] + \frac{d}{d x} [- 2]$

Step 3.2

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

$4 x^{3} - \frac{9 x^{2}}{2} + 5 (2 x) + \frac{d}{d x} [- 6 x] + \frac{d}{d x} [- 2]$

Step 3.3

Multiply $2$ by $5$ .

$4 x^{3} - \frac{9 x^{2}}{2} + 10 x + \frac{d}{d x} [- 6 x] + \frac{d}{d x} [- 2]$

Step 4

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

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Step 4.1

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

$4 x^{3} - \frac{9 x^{2}}{2} + 10 x - 6 \frac{d}{d x} [x] + \frac{d}{d x} [- 2]$

Step 4.2

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

$4 x^{3} - \frac{9 x^{2}}{2} + 10 x - 6 \cdot 1 + \frac{d}{d x} [- 2]$

Step 4.3

Multiply $- 6$ by $1$ .

$4 x^{3} - \frac{9 x^{2}}{2} + 10 x - 6 + \frac{d}{d x} [- 2]$

Step 5

Differentiate using the Constant Rule.

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Step 5.1

Since $- 2$ is constant with respect to $x$ , the derivative of $- 2$ with respect to $x$ is $0$ .

$4 x^{3} - \frac{9 x^{2}}{2} + 10 x - 6 + 0$

Step 5.2

Add $4 x^{3} - \frac{9 x^{2}}{2} + 10 x - 6$ and $0$ .

$4 x^{3} - \frac{9 x^{2}}{2} + 10 x - 6$