Calculus Examples

$z = 3 x^{2} - 6 y^{2} + x y + 5 y - 9$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (z) = \frac{d}{d x} (3 x^{2} - 6 y^{2} + x y + 5 y - 9)$

Step 2

The derivative of $z$ with respect to $x$ is $z'$ .

$z'$

Step 3

Differentiate the right side of the equation.

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Step 3.1

By the Sum Rule, the derivative of $3 x^{2} - 6 y^{2} + x y + 5 y - 9$ with respect to $x$ is $\frac{d}{d x} [3 x^{2}] + \frac{d}{d x} [- 6 y^{2}] + \frac{d}{d x} [x y] + \frac{d}{d x} [5 y] + \frac{d}{d x} [- 9]$ .

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

Step 3.2

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

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Step 3.2.1

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

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

Step 3.2.2

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

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

Step 3.2.3

Multiply $2$ by $3$ .

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

Step 3.3

Since $- 6 y^{2}$ is constant with respect to $x$ , the derivative of $- 6 y^{2}$ with respect to $x$ is $0$ .

$6 x + 0 + \frac{d}{d x} [x y] + \frac{d}{d x} [5 y] + \frac{d}{d x} [- 9]$

Step 3.4

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

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Step 3.4.1

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

$6 x + 0 + y \frac{d}{d x} [x] + \frac{d}{d x} [5 y] + \frac{d}{d x} [- 9]$

Step 3.4.2

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

$6 x + 0 + y \cdot 1 + \frac{d}{d x} [5 y] + \frac{d}{d x} [- 9]$

Step 3.4.3

Multiply $y$ by $1$ .

$6 x + 0 + y + \frac{d}{d x} [5 y] + \frac{d}{d x} [- 9]$

Step 3.5

Differentiate using the Constant Rule.

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Step 3.5.1

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

$6 x + 0 + y + 0 + \frac{d}{d x} [- 9]$

Step 3.5.2

Since $- 9$ is constant with respect to $x$ , the derivative of $- 9$ with respect to $x$ is $0$ .

$6 x + 0 + y + 0 + 0$

Step 3.6

Combine terms.

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Step 3.6.1

Add $6 x$ and $0$ .

$6 x + y + 0 + 0$

Step 3.6.2

Add $6 x + y$ and $0$ .

$6 x + y + 0$

Step 3.6.3

Add $6 x + y$ and $0$ .

$6 x + y$

Step 4

Reform the equation by setting the left side equal to the right side.

$z' = 6 x + y$

Step 5

Replace $z'$ with $\frac{d z}{d x}$ .

$\frac{d z}{d x} = 6 x + y$