Calculus Examples

$x^{2} - y + z^{2} = w$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d w} (x^{2} - y + z^{2}) = \frac{d}{d w} (w)$

Step 2

Differentiate the left side of the equation.

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Step 2.1

Differentiate.

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Step 2.1.1

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

$\frac{d}{d w} [x^{2}] + \frac{d}{d w} [- y] + \frac{d}{d w} [z^{2}]$

Step 2.1.2

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

$0 + \frac{d}{d w} [- y] + \frac{d}{d w} [z^{2}]$

Step 2.2

Evaluate $\frac{d}{d w} [- y]$ .

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Step 2.2.1

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

$0 - \frac{d}{d w} [y] + \frac{d}{d w} [z^{2}]$

Step 2.2.2

Rewrite $\frac{d}{d w} [y]$ as $y'$ .

$0 - y' + \frac{d}{d w} [z^{2}]$

Step 2.3

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

$0 - y' + 0$

Step 2.4

Combine terms.

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Step 2.4.1

Subtract $y'$ from $0$ .

$- y' + 0$

Step 2.4.2

Add $- y'$ and $0$ .

$- y'$

Step 3

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

$1$

Step 4

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

$- y' = 1$

Step 5

Divide each term in $- y' = 1$ by $- 1$ and simplify.

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Step 5.1

Divide each term in $- y' = 1$ by $- 1$ .

$\frac{- y'}{- 1} = \frac{1}{- 1}$

Step 5.2

Simplify the left side.

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Step 5.2.1

Dividing two negative values results in a positive value.

$\frac{y'}{1} = \frac{1}{- 1}$

Step 5.2.2

Divide $y'$ by $1$ .

$y' = \frac{1}{- 1}$

Step 5.3

Simplify the right side.

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Step 5.3.1

Divide $1$ by $- 1$ .

$y' = - 1$

Step 6

Replace $y'$ with $\frac{d y}{d w}$ .

$\frac{d y}{d w} = - 1$