Calculus Examples

$f (x) = 2 x^{2} + 2 y^{2} - \ln (1 - x^{2} - y^{2}) + 1$

Step 1

By the Sum Rule, the derivative of $2 x^{2} + 2 y^{2} - \ln (1 - x^{2} - y^{2}) + 1$ with respect to $x$ is $\frac{d}{d x} [2 x^{2}] + \frac{d}{d x} [2 y^{2}] + \frac{d}{d x} [- \ln (1 - x^{2} - y^{2})] + \frac{d}{d x} [1]$ .

$\frac{d}{d x} [2 x^{2}] + \frac{d}{d x} [2 y^{2}] + \frac{d}{d x} [- \ln (1 - x^{2} - y^{2})] + \frac{d}{d x} [1]$

Step 2

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

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Step 2.1

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

$2 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [2 y^{2}] + \frac{d}{d x} [- \ln (1 - x^{2} - y^{2})] + \frac{d}{d x} [1]$

Step 2.2

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

$2 (2 x) + \frac{d}{d x} [2 y^{2}] + \frac{d}{d x} [- \ln (1 - x^{2} - y^{2})] + \frac{d}{d x} [1]$

Step 2.3

Multiply $2$ by $2$ .

$4 x + \frac{d}{d x} [2 y^{2}] + \frac{d}{d x} [- \ln (1 - x^{2} - y^{2})] + \frac{d}{d x} [1]$

Step 3

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

$4 x + 0 + \frac{d}{d x} [- \ln (1 - x^{2} - y^{2})] + \frac{d}{d x} [1]$

Step 4

Evaluate $\frac{d}{d x} [- \ln (1 - x^{2} - y^{2})]$ .

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Step 4.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- \ln (1 - x^{2} - y^{2})$ with respect to $x$ is $- \frac{d}{d x} [\ln (1 - x^{2} - y^{2})]$ .

$4 x + 0 - \frac{d}{d x} [\ln (1 - x^{2} - y^{2})] + \frac{d}{d x} [1]$

Step 4.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \ln (x)$ and $g (x) = 1 - x^{2} - y^{2}$ .

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Step 4.2.1

To apply the Chain Rule, set $u$ as $1 - x^{2} - y^{2}$ .

$4 x + 0 - (\frac{d}{d u} [\ln (u)] \frac{d}{d x} [1 - x^{2} - y^{2}]) + \frac{d}{d x} [1]$

Step 4.2.2

The derivative of $\ln (u)$ with respect to $u$ is $\frac{1}{u}$ .

$4 x + 0 - (\frac{1}{u} \frac{d}{d x} [1 - x^{2} - y^{2}]) + \frac{d}{d x} [1]$

Step 4.2.3

Replace all occurrences of $u$ with $1 - x^{2} - y^{2}$ .

$4 x + 0 - (\frac{1}{1 - x^{2} - y^{2}} \frac{d}{d x} [1 - x^{2} - y^{2}]) + \frac{d}{d x} [1]$

Step 4.3

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

$4 x + 0 - (\frac{1}{1 - x^{2} - y^{2}} (\frac{d}{d x} [1] + \frac{d}{d x} [- x^{2}] + \frac{d}{d x} [- y^{2}])) + \frac{d}{d x} [1]$

Step 4.4

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

$4 x + 0 - (\frac{1}{1 - x^{2} - y^{2}} (0 + \frac{d}{d x} [- x^{2}] + \frac{d}{d x} [- y^{2}])) + \frac{d}{d x} [1]$

Step 4.5

Since $- 1$ is constant with respect to $x$ , the derivative of $- x^{2}$ with respect to $x$ is $- \frac{d}{d x} [x^{2}]$ .

$4 x + 0 - (\frac{1}{1 - x^{2} - y^{2}} (0 - \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- y^{2}])) + \frac{d}{d x} [1]$

Step 4.6

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

$4 x + 0 - (\frac{1}{1 - x^{2} - y^{2}} (0 - (2 x) + \frac{d}{d x} [- y^{2}])) + \frac{d}{d x} [1]$

Step 4.7

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

$4 x + 0 - (\frac{1}{1 - x^{2} - y^{2}} (0 - (2 x) + 0)) + \frac{d}{d x} [1]$

Step 4.8

Multiply $2$ by $- 1$ .

$4 x + 0 - (\frac{1}{1 - x^{2} - y^{2}} (0 - 2 x + 0)) + \frac{d}{d x} [1]$

Step 4.9

Subtract $2 x$ from $0$ .

$4 x + 0 - (\frac{1}{1 - x^{2} - y^{2}} (- 2 x + 0)) + \frac{d}{d x} [1]$

Step 4.10

Add $- 2 x$ and $0$ .

$4 x + 0 - (\frac{1}{1 - x^{2} - y^{2}} (- 2 x)) + \frac{d}{d x} [1]$

Step 4.11

Combine $- 2$ and $\frac{1}{1 - x^{2} - y^{2}}$ .

$4 x + 0 - (\frac{- 2}{1 - x^{2} - y^{2}} x) + \frac{d}{d x} [1]$

Step 4.12

Combine $\frac{- 2}{1 - x^{2} - y^{2}}$ and $x$ .

$4 x + 0 - \frac{- 2 x}{1 - x^{2} - y^{2}} + \frac{d}{d x} [1]$

Step 4.13

Move the negative in front of the fraction.

$4 x + 0 - - \frac{2 x}{1 - x^{2} - y^{2}} + \frac{d}{d x} [1]$

Step 4.14

Multiply $- 1$ by $- 1$ .

$4 x + 0 + 1 \frac{2 x}{1 - x^{2} - y^{2}} + \frac{d}{d x} [1]$

Step 4.15

Multiply $\frac{2 x}{1 - x^{2} - y^{2}}$ by $1$ .

$4 x + 0 + \frac{2 x}{1 - x^{2} - y^{2}} + \frac{d}{d x} [1]$

Step 5

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

$4 x + 0 + \frac{2 x}{1 - x^{2} - y^{2}} + 0$

Step 6

Simplify.

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Step 6.1

Combine terms.

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Step 6.1.1

Add $4 x$ and $0$ .

$4 x + \frac{2 x}{1 - x^{2} - y^{2}} + 0$

Step 6.1.2

To write $4 x$ as a fraction with a common denominator, multiply by $\frac{1 - x^{2} - y^{2}}{1 - x^{2} - y^{2}}$ .

$\frac{4 x (1 - x^{2} - y^{2})}{1 - x^{2} - y^{2}} + \frac{2 x}{1 - x^{2} - y^{2}} + 0$

Step 6.1.3

Combine the numerators over the common denominator.

$\frac{4 x (1 - x^{2} - y^{2}) + 2 x}{1 - x^{2} - y^{2}} + 0$

Step 6.1.4

Add $\frac{4 x (1 - x^{2} - y^{2}) + 2 x}{1 - x^{2} - y^{2}}$ and $0$ .

$\frac{4 x (1 - x^{2} - y^{2}) + 2 x}{1 - x^{2} - y^{2}}$

Step 6.2

Reorder terms.

$\frac{4 x (1 - x^{2} - y^{2}) + 2 x}{- x^{2} - y^{2} + 1}$