Calculus Examples

$f (x) = 9 x - x^{3} + 6 x^{2} - 15 x + 1000 y^{2} + y + 1$

Step 1

By the Sum Rule, the derivative of $9 x - x^{3} + 6 x^{2} - 15 x + 1000 y^{2} + y + 1$ with respect to $x$ is $\frac{d}{d x} [9 x] + \frac{d}{d x} [- x^{3}] + \frac{d}{d x} [6 x^{2}] + \frac{d}{d x} [- 15 x] + \frac{d}{d x} [1000 y^{2}] + \frac{d}{d x} [y] + \frac{d}{d x} [1]$ .

$\frac{d}{d x} [9 x] + \frac{d}{d x} [- x^{3}] + \frac{d}{d x} [6 x^{2}] + \frac{d}{d x} [- 15 x] + \frac{d}{d x} [1000 y^{2}] + \frac{d}{d x} [y] + \frac{d}{d x} [1]$

Step 2

Evaluate $\frac{d}{d x} [9 x]$ .

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Step 2.1

Since $9$ is constant with respect to $x$ , the derivative of $9 x$ with respect to $x$ is $9 \frac{d}{d x} [x]$ .

$9 \frac{d}{d x} [x] + \frac{d}{d x} [- x^{3}] + \frac{d}{d x} [6 x^{2}] + \frac{d}{d x} [- 15 x] + \frac{d}{d x} [1000 y^{2}] + \frac{d}{d x} [y] + \frac{d}{d x} [1]$

Step 2.2

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

$9 \cdot 1 + \frac{d}{d x} [- x^{3}] + \frac{d}{d x} [6 x^{2}] + \frac{d}{d x} [- 15 x] + \frac{d}{d x} [1000 y^{2}] + \frac{d}{d x} [y] + \frac{d}{d x} [1]$

Step 2.3

Multiply $9$ by $1$ .

$9 + \frac{d}{d x} [- x^{3}] + \frac{d}{d x} [6 x^{2}] + \frac{d}{d x} [- 15 x] + \frac{d}{d x} [1000 y^{2}] + \frac{d}{d x} [y] + \frac{d}{d x} [1]$

Step 3

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

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Step 3.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- x^{3}$ with respect to $x$ is $- \frac{d}{d x} [x^{3}]$ .

$9 - \frac{d}{d x} [x^{3}] + \frac{d}{d x} [6 x^{2}] + \frac{d}{d x} [- 15 x] + \frac{d}{d x} [1000 y^{2}] + \frac{d}{d x} [y] + \frac{d}{d x} [1]$

Step 3.2

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

$9 - (3 x^{2}) + \frac{d}{d x} [6 x^{2}] + \frac{d}{d x} [- 15 x] + \frac{d}{d x} [1000 y^{2}] + \frac{d}{d x} [y] + \frac{d}{d x} [1]$

Step 3.3

Multiply $3$ by $- 1$ .

$9 - 3 x^{2} + \frac{d}{d x} [6 x^{2}] + \frac{d}{d x} [- 15 x] + \frac{d}{d x} [1000 y^{2}] + \frac{d}{d x} [y] + \frac{d}{d x} [1]$

Step 4

Evaluate $\frac{d}{d x} [6 x^{2}]$ .

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Step 4.1

Since $6$ is constant with respect to $x$ , the derivative of $6 x^{2}$ with respect to $x$ is $6 \frac{d}{d x} [x^{2}]$ .

$9 - 3 x^{2} + 6 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 15 x] + \frac{d}{d x} [1000 y^{2}] + \frac{d}{d x} [y] + \frac{d}{d x} [1]$

Step 4.2

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

$9 - 3 x^{2} + 6 (2 x) + \frac{d}{d x} [- 15 x] + \frac{d}{d x} [1000 y^{2}] + \frac{d}{d x} [y] + \frac{d}{d x} [1]$

Step 4.3

Multiply $2$ by $6$ .

$9 - 3 x^{2} + 12 x + \frac{d}{d x} [- 15 x] + \frac{d}{d x} [1000 y^{2}] + \frac{d}{d x} [y] + \frac{d}{d x} [1]$

Step 5

Evaluate $\frac{d}{d x} [- 15 x]$ .

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Step 5.1

Since $- 15$ is constant with respect to $x$ , the derivative of $- 15 x$ with respect to $x$ is $- 15 \frac{d}{d x} [x]$ .

$9 - 3 x^{2} + 12 x - 15 \frac{d}{d x} [x] + \frac{d}{d x} [1000 y^{2}] + \frac{d}{d x} [y] + \frac{d}{d x} [1]$

Step 5.2

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

$9 - 3 x^{2} + 12 x - 15 \cdot 1 + \frac{d}{d x} [1000 y^{2}] + \frac{d}{d x} [y] + \frac{d}{d x} [1]$

Step 5.3

Multiply $- 15$ by $1$ .

$9 - 3 x^{2} + 12 x - 15 + \frac{d}{d x} [1000 y^{2}] + \frac{d}{d x} [y] + \frac{d}{d x} [1]$

Step 6

Differentiate using the Constant Rule.

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Step 6.1

Since $1000 y^{2}$ is constant with respect to $x$ , the derivative of $1000 y^{2}$ with respect to $x$ is $0$ .

$9 - 3 x^{2} + 12 x - 15 + 0 + \frac{d}{d x} [y] + \frac{d}{d x} [1]$

Step 6.2

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

$9 - 3 x^{2} + 12 x - 15 + 0 + 0 + \frac{d}{d x} [1]$

Step 6.3

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$9 - 3 x^{2} + 12 x - 15 + 0 + 0 + 0$

Step 7

Combine terms.

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Step 7.1

Subtract $15$ from $9$ .

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

Step 7.2

Add $- 3 x^{2} + 12 x - 6$ and $0$ .

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

Step 7.3

Add $- 3 x^{2} + 12 x - 6$ and $0$ .

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

Step 7.4

Add $- 3 x^{2} + 12 x - 6$ and $0$ .

$- 3 x^{2} + 12 x - 6$