Calculus Examples

$2 x^{0.5} + 6 z^{2} - 4 x^{- 0.5} - 8 t^{4} + 2 z + 33$

Step 1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{d}{d z} [2 x^{0.5} + 6 z^{2} - 4 \frac{1}{x^{0.5}} - 8 t^{4} + 2 z + 33]$

Step 2

Differentiate.

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Step 2.1

By the Sum Rule, the derivative of $2 x^{0.5} + 6 z^{2} - 4 \frac{1}{x^{0.5}} - 8 t^{4} + 2 z + 33$ with respect to $z$ is $\frac{d}{d z} [2 x^{0.5}] + \frac{d}{d z} [6 z^{2}] + \frac{d}{d z} [- 4 \frac{1}{x^{0.5}}] + \frac{d}{d z} [- 8 t^{4}] + \frac{d}{d z} [2 z] + \frac{d}{d z} [33]$ .

$\frac{d}{d z} [2 x^{0.5}] + \frac{d}{d z} [6 z^{2}] + \frac{d}{d z} [- 4 \frac{1}{x^{0.5}}] + \frac{d}{d z} [- 8 t^{4}] + \frac{d}{d z} [2 z] + \frac{d}{d z} [33]$

Step 2.2

Since $2 x^{0.5}$ is constant with respect to $z$ , the derivative of $2 x^{0.5}$ with respect to $z$ is $0$ .

$0 + \frac{d}{d z} [6 z^{2}] + \frac{d}{d z} [- 4 \frac{1}{x^{0.5}}] + \frac{d}{d z} [- 8 t^{4}] + \frac{d}{d z} [2 z] + \frac{d}{d z} [33]$

Step 3

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

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Step 3.1

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

$0 + 6 \frac{d}{d z} [z^{2}] + \frac{d}{d z} [- 4 \frac{1}{x^{0.5}}] + \frac{d}{d z} [- 8 t^{4}] + \frac{d}{d z} [2 z] + \frac{d}{d z} [33]$

Step 3.2

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

$0 + 6 (2 z) + \frac{d}{d z} [- 4 \frac{1}{x^{0.5}}] + \frac{d}{d z} [- 8 t^{4}] + \frac{d}{d z} [2 z] + \frac{d}{d z} [33]$

Step 3.3

Multiply $2$ by $6$ .

$0 + 12 z + \frac{d}{d z} [- 4 \frac{1}{x^{0.5}}] + \frac{d}{d z} [- 8 t^{4}] + \frac{d}{d z} [2 z] + \frac{d}{d z} [33]$

Step 4

Evaluate $\frac{d}{d z} [- 4 \frac{1}{x^{0.5}}]$ .

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Step 4.1

Combine $- 4$ and $\frac{1}{x^{0.5}}$ .

$0 + 12 z + \frac{d}{d z} [\frac{- 4}{x^{0.5}}] + \frac{d}{d z} [- 8 t^{4}] + \frac{d}{d z} [2 z] + \frac{d}{d z} [33]$

Step 4.2

Move the negative in front of the fraction.

$0 + 12 z + \frac{d}{d z} [- \frac{4}{x^{0.5}}] + \frac{d}{d z} [- 8 t^{4}] + \frac{d}{d z} [2 z] + \frac{d}{d z} [33]$

Step 4.3

Since $- \frac{4}{x^{0.5}}$ is constant with respect to $z$ , the derivative of $- \frac{4}{x^{0.5}}$ with respect to $z$ is $0$ .

$0 + 12 z + 0 + \frac{d}{d z} [- 8 t^{4}] + \frac{d}{d z} [2 z] + \frac{d}{d z} [33]$

Step 5

Since $- 8 t^{4}$ is constant with respect to $z$ , the derivative of $- 8 t^{4}$ with respect to $z$ is $0$ .

$0 + 12 z + 0 + 0 + \frac{d}{d z} [2 z] + \frac{d}{d z} [33]$

Step 6

Evaluate $\frac{d}{d z} [2 z]$ .

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Step 6.1

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

$0 + 12 z + 0 + 0 + 2 \frac{d}{d z} [z] + \frac{d}{d z} [33]$

Step 6.2

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

$0 + 12 z + 0 + 0 + 2 \cdot 1 + \frac{d}{d z} [33]$

Step 6.3

Multiply $2$ by $1$ .

$0 + 12 z + 0 + 0 + 2 + \frac{d}{d z} [33]$

Step 7

Since $33$ is constant with respect to $z$ , the derivative of $33$ with respect to $z$ is $0$ .

$0 + 12 z + 0 + 0 + 2 + 0$

Step 8

Combine terms.

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Step 8.1

Add $0$ and $12 z$ .

$12 z + 0 + 0 + 2 + 0$

Step 8.2

Add $12 z$ and $0$ .

$12 z + 0 + 2 + 0$

Step 8.3

Add $12 z$ and $0$ .

$12 z + 2 + 0$

Step 8.4

Add $12 z + 2$ and $0$ .

$12 z + 2$