Calculus Examples

$N (t) = - \frac{20000}{{(1 + 0.2 t)}^{\frac{1}{2}}} + c$

Step 1

By the Sum Rule, the derivative of $- \frac{20000}{{(1 + 0.2 t)}^{\frac{1}{2}}} + c$ with respect to $t$ is $\frac{d}{d t} [- \frac{20000}{{(1 + 0.2 t)}^{\frac{1}{2}}}] + \frac{d}{d t} [c]$ .

$\frac{d}{d t} [- \frac{20000}{{(1 + 0.2 t)}^{\frac{1}{2}}}] + \frac{d}{d t} [c]$

Step 2

Evaluate $\frac{d}{d t} [- \frac{20000}{{(1 + 0.2 t)}^{\frac{1}{2}}}]$ .

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Step 2.1

Since $- 20000$ is constant with respect to $t$ , the derivative of $- \frac{20000}{{(1 + 0.2 t)}^{\frac{1}{2}}}$ with respect to $t$ is $- 20000 \frac{d}{d t} [\frac{1}{{(1 + 0.2 t)}^{\frac{1}{2}}}]$ .

$- 20000 \frac{d}{d t} [\frac{1}{{(1 + 0.2 t)}^{\frac{1}{2}}}] + \frac{d}{d t} [c]$

Step 2.2

Rewrite $\frac{1}{{(1 + 0.2 t)}^{\frac{1}{2}}}$ as ${({(1 + 0.2 t)}^{\frac{1}{2}})}^{- 1}$ .

$- 20000 \frac{d}{d t} [{({(1 + 0.2 t)}^{\frac{1}{2}})}^{- 1}] + \frac{d}{d t} [c]$

Step 2.3

Differentiate using the chain rule, which states that $\frac{d}{d t} [f (g (t))]$ is $f' (g (t)) g' (t)$ where $f (t) = t^{- 1}$ and $g (t) = {(1 + 0.2 t)}^{\frac{1}{2}}$ .

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Step 2.3.1

To apply the Chain Rule, set $u_{1}$ as ${(1 + 0.2 t)}^{\frac{1}{2}}$ .

$- 20000 (\frac{d}{d u_{1}} [{u_{1}}^{- 1}] \frac{d}{d t} [{(1 + 0.2 t)}^{\frac{1}{2}}]) + \frac{d}{d t} [c]$

Step 2.3.2

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

$- 20000 (- {u_{1}}^{- 2} \frac{d}{d t} [{(1 + 0.2 t)}^{\frac{1}{2}}]) + \frac{d}{d t} [c]$

Step 2.3.3

Replace all occurrences of $u_{1}$ with ${(1 + 0.2 t)}^{\frac{1}{2}}$ .

$- 20000 (- {({(1 + 0.2 t)}^{\frac{1}{2}})}^{- 2} \frac{d}{d t} [{(1 + 0.2 t)}^{\frac{1}{2}}]) + \frac{d}{d t} [c]$

Step 2.4

Differentiate using the chain rule, which states that $\frac{d}{d t} [f (g (t))]$ is $f' (g (t)) g' (t)$ where $f (t) = t^{\frac{1}{2}}$ and $g (t) = 1 + 0.2 t$ .

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Step 2.4.1

To apply the Chain Rule, set $u_{2}$ as $1 + 0.2 t$ .

$- 20000 (- {({(1 + 0.2 t)}^{\frac{1}{2}})}^{- 2} (\frac{d}{d u_{2}} [{u_{2}}^{\frac{1}{2}}] \frac{d}{d t} [1 + 0.2 t])) + \frac{d}{d t} [c]$

Step 2.4.2

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

$- 20000 (- {({(1 + 0.2 t)}^{\frac{1}{2}})}^{- 2} (\frac{1}{2} {u_{2}}^{\frac{1}{2} - 1} \frac{d}{d t} [1 + 0.2 t])) + \frac{d}{d t} [c]$

Step 2.4.3

Replace all occurrences of $u_{2}$ with $1 + 0.2 t$ .

$- 20000 (- {({(1 + 0.2 t)}^{\frac{1}{2}})}^{- 2} (\frac{1}{2} {(1 + 0.2 t)}^{\frac{1}{2} - 1} \frac{d}{d t} [1 + 0.2 t])) + \frac{d}{d t} [c]$

Step 2.5

By the Sum Rule, the derivative of $1 + 0.2 t$ with respect to $t$ is $\frac{d}{d t} [1] + \frac{d}{d t} [0.2 t]$ .

$- 20000 (- {({(1 + 0.2 t)}^{\frac{1}{2}})}^{- 2} (\frac{1}{2} {(1 + 0.2 t)}^{\frac{1}{2} - 1} (\frac{d}{d t} [1] + \frac{d}{d t} [0.2 t]))) + \frac{d}{d t} [c]$

Step 2.6

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

$- 20000 (- {({(1 + 0.2 t)}^{\frac{1}{2}})}^{- 2} (\frac{1}{2} {(1 + 0.2 t)}^{\frac{1}{2} - 1} (0 + \frac{d}{d t} [0.2 t]))) + \frac{d}{d t} [c]$

Step 2.7

Since $0.2$ is constant with respect to $t$ , the derivative of $0.2 t$ with respect to $t$ is $0.2 \frac{d}{d t} [t]$ .

$- 20000 (- {({(1 + 0.2 t)}^{\frac{1}{2}})}^{- 2} (\frac{1}{2} {(1 + 0.2 t)}^{\frac{1}{2} - 1} (0 + 0.2 \frac{d}{d t} [t]))) + \frac{d}{d t} [c]$

Step 2.8

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

$- 20000 (- {({(1 + 0.2 t)}^{\frac{1}{2}})}^{- 2} (\frac{1}{2} {(1 + 0.2 t)}^{\frac{1}{2} - 1} (0 + 0.2 \cdot 1))) + \frac{d}{d t} [c]$

Step 2.9

Multiply the exponents in ${({(1 + 0.2 t)}^{\frac{1}{2}})}^{- 2}$ .

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Step 2.9.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$- 20000 (- {(1 + 0.2 t)}^{\frac{1}{2} \cdot - 2} (\frac{1}{2} {(1 + 0.2 t)}^{\frac{1}{2} - 1} (0 + 0.2 \cdot 1))) + \frac{d}{d t} [c]$

Step 2.9.2

Cancel the common factor of $2$ .

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Step 2.9.2.1

Factor $2$ out of $- 2$ .

$- 20000 (- {(1 + 0.2 t)}^{\frac{1}{2} \cdot (2 (- 1))} (\frac{1}{2} {(1 + 0.2 t)}^{\frac{1}{2} - 1} (0 + 0.2 \cdot 1))) + \frac{d}{d t} [c]$

Step 2.9.2.2

Cancel the common factor.

$- 20000 (- {(1 + 0.2 t)}^{\frac{1}{2} \cdot (2 \cdot - 1)} (\frac{1}{2} {(1 + 0.2 t)}^{\frac{1}{2} - 1} (0 + 0.2 \cdot 1))) + \frac{d}{d t} [c]$

Step 2.9.2.3

Rewrite the expression.

$- 20000 (- {(1 + 0.2 t)}^{- 1} (\frac{1}{2} {(1 + 0.2 t)}^{\frac{1}{2} - 1} (0 + 0.2 \cdot 1))) + \frac{d}{d t} [c]$

Step 2.10

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

$- 20000 (- {(1 + 0.2 t)}^{- 1} (\frac{1}{2} {(1 + 0.2 t)}^{\frac{1}{2} - 1 \cdot \frac{2}{2}} (0 + 0.2 \cdot 1))) + \frac{d}{d t} [c]$

Step 2.11

Combine $- 1$ and $\frac{2}{2}$ .

$- 20000 (- {(1 + 0.2 t)}^{- 1} (\frac{1}{2} {(1 + 0.2 t)}^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}} (0 + 0.2 \cdot 1))) + \frac{d}{d t} [c]$

Step 2.12

Combine the numerators over the common denominator.

$- 20000 (- {(1 + 0.2 t)}^{- 1} (\frac{1}{2} {(1 + 0.2 t)}^{\frac{1 - 1 \cdot 2}{2}} (0 + 0.2 \cdot 1))) + \frac{d}{d t} [c]$

Step 2.13

Simplify the numerator.

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Step 2.13.1

Multiply $- 1$ by $2$ .

$- 20000 (- {(1 + 0.2 t)}^{- 1} (\frac{1}{2} {(1 + 0.2 t)}^{\frac{1 - 2}{2}} (0 + 0.2 \cdot 1))) + \frac{d}{d t} [c]$

Step 2.13.2

Subtract $2$ from $1$ .

$- 20000 (- {(1 + 0.2 t)}^{- 1} (\frac{1}{2} {(1 + 0.2 t)}^{\frac{- 1}{2}} (0 + 0.2 \cdot 1))) + \frac{d}{d t} [c]$

Step 2.14

Move the negative in front of the fraction.

$- 20000 (- {(1 + 0.2 t)}^{- 1} (\frac{1}{2} {(1 + 0.2 t)}^{- \frac{1}{2}} (0 + 0.2 \cdot 1))) + \frac{d}{d t} [c]$

Step 2.15

Multiply $0.2$ by $1$ .

$- 20000 (- {(1 + 0.2 t)}^{- 1} (\frac{1}{2} {(1 + 0.2 t)}^{- \frac{1}{2}} (0 + 0.2))) + \frac{d}{d t} [c]$

Step 2.16

Add $0$ and $0.2$ .

$- 20000 (- {(1 + 0.2 t)}^{- 1} (\frac{1}{2} {(1 + 0.2 t)}^{- \frac{1}{2}} \cdot 0.2)) + \frac{d}{d t} [c]$

Step 2.17

Combine $\frac{1}{2}$ and ${(1 + 0.2 t)}^{- \frac{1}{2}}$ .

$- 20000 (- {(1 + 0.2 t)}^{- 1} (\frac{{(1 + 0.2 t)}^{- \frac{1}{2}}}{2} \cdot 0.2)) + \frac{d}{d t} [c]$

Step 2.18

Combine $\frac{{(1 + 0.2 t)}^{- \frac{1}{2}}}{2}$ and $0.2$ .

$- 20000 (- {(1 + 0.2 t)}^{- 1} \frac{{(1 + 0.2 t)}^{- \frac{1}{2}} \cdot 0.2}{2}) + \frac{d}{d t} [c]$

Step 2.19

Move $0.2$ to the left of ${(1 + 0.2 t)}^{- \frac{1}{2}}$ .

$- 20000 (- {(1 + 0.2 t)}^{- 1} \frac{0.2 \cdot {(1 + 0.2 t)}^{- \frac{1}{2}}}{2}) + \frac{d}{d t} [c]$

Step 2.20

Move ${(1 + 0.2 t)}^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$- 20000 (- {(1 + 0.2 t)}^{- 1} \frac{0.2}{2 {(1 + 0.2 t)}^{\frac{1}{2}}}) + \frac{d}{d t} [c]$

Step 2.21

Combine $\frac{0.2}{2 {(1 + 0.2 t)}^{\frac{1}{2}}}$ and ${(1 + 0.2 t)}^{- 1}$ .

$- 20000 (- \frac{0.2 {(1 + 0.2 t)}^{- 1}}{2 {(1 + 0.2 t)}^{\frac{1}{2}}}) + \frac{d}{d t} [c]$

Step 2.22

Move ${(1 + 0.2 t)}^{- 1}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$- 20000 (- \frac{0.2}{2 {(1 + 0.2 t)}^{\frac{1}{2}} (1 + 0.2 t)}) + \frac{d}{d t} [c]$

Step 2.23

Multiply ${(1 + 0.2 t)}^{\frac{1}{2}}$ by $1 + 0.2 t$ by adding the exponents.

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Step 2.23.1

Move $1 + 0.2 t$ .

$- 20000 (- \frac{0.2}{2 ((1 + 0.2 t) {(1 + 0.2 t)}^{\frac{1}{2}})}) + \frac{d}{d t} [c]$

Step 2.23.2

Multiply $1 + 0.2 t$ by ${(1 + 0.2 t)}^{\frac{1}{2}}$ .

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Step 2.23.2.1

Raise $1 + 0.2 t$ to the power of $1$ .

$- 20000 (- \frac{0.2}{2 ({(1 + 0.2 t)}^{1} {(1 + 0.2 t)}^{\frac{1}{2}})}) + \frac{d}{d t} [c]$

Step 2.23.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$- 20000 (- \frac{0.2}{2 {(1 + 0.2 t)}^{1 + \frac{1}{2}}}) + \frac{d}{d t} [c]$

Step 2.23.3

Write $1$ as a fraction with a common denominator.

$- 20000 (- \frac{0.2}{2 {(1 + 0.2 t)}^{\frac{2}{2} + \frac{1}{2}}}) + \frac{d}{d t} [c]$

Step 2.23.4

Combine the numerators over the common denominator.

$- 20000 (- \frac{0.2}{2 {(1 + 0.2 t)}^{\frac{2 + 1}{2}}}) + \frac{d}{d t} [c]$

Step 2.23.5

Add $2$ and $1$ .

$- 20000 (- \frac{0.2}{2 {(1 + 0.2 t)}^{\frac{3}{2}}}) + \frac{d}{d t} [c]$

Step 2.24

Multiply $- 1$ by $- 20000$ .

$20000 \frac{0.2}{2 {(1 + 0.2 t)}^{\frac{3}{2}}} + \frac{d}{d t} [c]$

Step 2.25

Combine $20000$ and $\frac{0.2}{2 {(1 + 0.2 t)}^{\frac{3}{2}}}$ .

$\frac{20000 \cdot 0.2}{2 {(1 + 0.2 t)}^{\frac{3}{2}}} + \frac{d}{d t} [c]$

Step 2.26

Multiply $20000$ by $0.2$ .

$\frac{4000}{2 {(1 + 0.2 t)}^{\frac{3}{2}}} + \frac{d}{d t} [c]$

Step 2.27

Factor $2$ out of $4000$ .

$\frac{2 \cdot 2000}{2 {(1 + 0.2 t)}^{\frac{3}{2}}} + \frac{d}{d t} [c]$

Step 2.28

Cancel the common factors.

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Step 2.28.1

Factor $2$ out of $2 {(1 + 0.2 t)}^{\frac{3}{2}}$ .

$\frac{2 \cdot 2000}{2 ({(1 + 0.2 t)}^{\frac{3}{2}})} + \frac{d}{d t} [c]$

Step 2.28.2

Cancel the common factor.

$\frac{2 \cdot 2000}{2 {(1 + 0.2 t)}^{\frac{3}{2}}} + \frac{d}{d t} [c]$

Step 2.28.3

Rewrite the expression.

$\frac{2000}{{(1 + 0.2 t)}^{\frac{3}{2}}} + \frac{d}{d t} [c]$

Step 3

Since $c$ is constant with respect to $t$ , the derivative of $c$ with respect to $t$ is $0$ .

$\frac{2000}{{(1 + 0.2 t)}^{\frac{3}{2}}} + 0$

Step 4

Simplify.

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Step 4.1

Add $\frac{2000}{{(1 + 0.2 t)}^{\frac{3}{2}}}$ and $0$ .

$\frac{2000}{{(1 + 0.2 t)}^{\frac{3}{2}}}$

Step 4.2

Reorder terms.

$\frac{2000}{{(0.2 t + 1)}^{\frac{3}{2}}}$