Calculus Examples

$300 \sqrt[3]{x^{2}} \sqrt[4]{y} - 100 x - 150 y + 2021$

Step 1

By the Sum Rule, the derivative of $300 \sqrt[3]{x^{2}} \sqrt[4]{y} - 100 x - 150 y + 2021$ with respect to $x$ is $\frac{d}{d x} [300 \sqrt[3]{x^{2}} \sqrt[4]{y}] + \frac{d}{d x} [- 100 x] + \frac{d}{d x} [- 150 y] + \frac{d}{d x} [2021]$ .

$\frac{d}{d x} [300 \sqrt[3]{x^{2}} \sqrt[4]{y}] + \frac{d}{d x} [- 100 x] + \frac{d}{d x} [- 150 y] + \frac{d}{d x} [2021]$

Step 2

Evaluate $\frac{d}{d x} [300 \sqrt[3]{x^{2}} \sqrt[4]{y}]$ .

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Step 2.1

Rewrite the expression using the least common index of $12$ .

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Step 2.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[4]{y}$ as $y^{\frac{1}{4}}$ .

$\frac{d}{d x} [300 (y^{\frac{1}{4}} \sqrt[3]{x^{2}})] + \frac{d}{d x} [- 100 x] + \frac{d}{d x} [- 150 y] + \frac{d}{d x} [2021]$

Step 2.1.2

Rewrite $y^{\frac{1}{4}}$ as $y^{\frac{3}{12}}$ .

$\frac{d}{d x} [300 (y^{\frac{3}{12}} \sqrt[3]{x^{2}})] + \frac{d}{d x} [- 100 x] + \frac{d}{d x} [- 150 y] + \frac{d}{d x} [2021]$

Step 2.1.3

Rewrite $y^{\frac{3}{12}}$ as $\sqrt[12]{y^{3}}$ .

$\frac{d}{d x} [300 (\sqrt[12]{y^{3}} \sqrt[3]{x^{2}})] + \frac{d}{d x} [- 100 x] + \frac{d}{d x} [- 150 y] + \frac{d}{d x} [2021]$

Step 2.1.4

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{x^{2}}$ as $x^{\frac{2}{3}}$ .

$\frac{d}{d x} [300 (\sqrt[12]{y^{3}} x^{\frac{2}{3}})] + \frac{d}{d x} [- 100 x] + \frac{d}{d x} [- 150 y] + \frac{d}{d x} [2021]$

Step 2.1.5

Rewrite $x^{\frac{2}{3}}$ as $x^{\frac{8}{12}}$ .

$\frac{d}{d x} [300 (\sqrt[12]{y^{3}} x^{\frac{8}{12}})] + \frac{d}{d x} [- 100 x] + \frac{d}{d x} [- 150 y] + \frac{d}{d x} [2021]$

Step 2.1.6

Rewrite $x^{\frac{8}{12}}$ as $\sqrt[12]{x^{8}}$ .

$\frac{d}{d x} [300 (\sqrt[12]{y^{3}} \sqrt[12]{x^{8}})] + \frac{d}{d x} [- 100 x] + \frac{d}{d x} [- 150 y] + \frac{d}{d x} [2021]$

Step 2.2

Combine using the product rule for radicals.

$\frac{d}{d x} [300 \sqrt[12]{y^{3} x^{8}}] + \frac{d}{d x} [- 100 x] + \frac{d}{d x} [- 150 y] + \frac{d}{d x} [2021]$

Step 2.3

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[12]{y^{3} x^{8}}$ as ${(y^{3} x^{8})}^{\frac{1}{12}}$ .

$\frac{d}{d x} [300 {(y^{3} x^{8})}^{\frac{1}{12}}] + \frac{d}{d x} [- 100 x] + \frac{d}{d x} [- 150 y] + \frac{d}{d x} [2021]$

Step 2.4

Since $300$ is constant with respect to $x$ , the derivative of $300 {(y^{3} x^{8})}^{\frac{1}{12}}$ with respect to $x$ is $300 \frac{d}{d x} [{(y^{3} x^{8})}^{\frac{1}{12}}]$ .

$300 \frac{d}{d x} [{(y^{3} x^{8})}^{\frac{1}{12}}] + \frac{d}{d x} [- 100 x] + \frac{d}{d x} [- 150 y] + \frac{d}{d x} [2021]$

Step 2.5

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{\frac{1}{12}}$ and $g (x) = y^{3} x^{8}$ .

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Step 2.5.1

To apply the Chain Rule, set $u$ as $y^{3} x^{8}$ .

$300 (\frac{d}{d u} [u^{\frac{1}{12}}] \frac{d}{d x} [y^{3} x^{8}]) + \frac{d}{d x} [- 100 x] + \frac{d}{d x} [- 150 y] + \frac{d}{d x} [2021]$

Step 2.5.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = \frac{1}{12}$ .

$300 (\frac{1}{12} u^{\frac{1}{12} - 1} \frac{d}{d x} [y^{3} x^{8}]) + \frac{d}{d x} [- 100 x] + \frac{d}{d x} [- 150 y] + \frac{d}{d x} [2021]$

Step 2.5.3

Replace all occurrences of $u$ with $y^{3} x^{8}$ .

$300 (\frac{1}{12} {(y^{3} x^{8})}^{\frac{1}{12} - 1} \frac{d}{d x} [y^{3} x^{8}]) + \frac{d}{d x} [- 100 x] + \frac{d}{d x} [- 150 y] + \frac{d}{d x} [2021]$

Step 2.6

Since $y^{3}$ is constant with respect to $x$ , the derivative of $y^{3} x^{8}$ with respect to $x$ is $y^{3} \frac{d}{d x} [x^{8}]$ .

$300 (\frac{1}{12} {(y^{3} x^{8})}^{\frac{1}{12} - 1} (y^{3} \frac{d}{d x} [x^{8}])) + \frac{d}{d x} [- 100 x] + \frac{d}{d x} [- 150 y] + \frac{d}{d x} [2021]$

Step 2.7

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 8$ .

$300 (\frac{1}{12} {(y^{3} x^{8})}^{\frac{1}{12} - 1} (y^{3} (8 x^{7}))) + \frac{d}{d x} [- 100 x] + \frac{d}{d x} [- 150 y] + \frac{d}{d x} [2021]$

Step 2.8

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{12}{12}$ .

$300 (\frac{1}{12} {(y^{3} x^{8})}^{\frac{1}{12} - 1 \cdot \frac{12}{12}} (y^{3} (8 x^{7}))) + \frac{d}{d x} [- 100 x] + \frac{d}{d x} [- 150 y] + \frac{d}{d x} [2021]$

Step 2.9

Combine $- 1$ and $\frac{12}{12}$ .

$300 (\frac{1}{12} {(y^{3} x^{8})}^{\frac{1}{12} + \frac{- 1 \cdot 12}{12}} (y^{3} (8 x^{7}))) + \frac{d}{d x} [- 100 x] + \frac{d}{d x} [- 150 y] + \frac{d}{d x} [2021]$

Step 2.10

Combine the numerators over the common denominator.

$300 (\frac{1}{12} {(y^{3} x^{8})}^{\frac{1 - 1 \cdot 12}{12}} (y^{3} (8 x^{7}))) + \frac{d}{d x} [- 100 x] + \frac{d}{d x} [- 150 y] + \frac{d}{d x} [2021]$

Step 2.11

Simplify the numerator.

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Step 2.11.1

Multiply $- 1$ by $12$ .

$300 (\frac{1}{12} {(y^{3} x^{8})}^{\frac{1 - 12}{12}} (y^{3} (8 x^{7}))) + \frac{d}{d x} [- 100 x] + \frac{d}{d x} [- 150 y] + \frac{d}{d x} [2021]$

Step 2.11.2

Subtract $12$ from $1$ .

$300 (\frac{1}{12} {(y^{3} x^{8})}^{\frac{- 11}{12}} (y^{3} (8 x^{7}))) + \frac{d}{d x} [- 100 x] + \frac{d}{d x} [- 150 y] + \frac{d}{d x} [2021]$

Step 2.12

Move the negative in front of the fraction.

$300 (\frac{1}{12} {(y^{3} x^{8})}^{- \frac{11}{12}} (y^{3} (8 x^{7}))) + \frac{d}{d x} [- 100 x] + \frac{d}{d x} [- 150 y] + \frac{d}{d x} [2021]$

Step 2.13

Move $8$ to the left of $y^{3}$ .

$300 (\frac{1}{12} {(y^{3} x^{8})}^{- \frac{11}{12}} (8 \cdot y^{3} x^{7})) + \frac{d}{d x} [- 100 x] + \frac{d}{d x} [- 150 y] + \frac{d}{d x} [2021]$

Step 2.14

Combine $\frac{1}{12}$ and ${(y^{3} x^{8})}^{- \frac{11}{12}}$ .

$300 (\frac{{(y^{3} x^{8})}^{- \frac{11}{12}}}{12} (8 y^{3} x^{7})) + \frac{d}{d x} [- 100 x] + \frac{d}{d x} [- 150 y] + \frac{d}{d x} [2021]$

Step 2.15

Combine $8$ and $\frac{{(y^{3} x^{8})}^{- \frac{11}{12}}}{12}$ .

$300 (\frac{8 {(y^{3} x^{8})}^{- \frac{11}{12}}}{12} (y^{3} x^{7})) + \frac{d}{d x} [- 100 x] + \frac{d}{d x} [- 150 y] + \frac{d}{d x} [2021]$

Step 2.16

Combine $y^{3}$ and $\frac{8 {(y^{3} x^{8})}^{- \frac{11}{12}}}{12}$ .

$300 (\frac{y^{3} (8 {(y^{3} x^{8})}^{- \frac{11}{12}})}{12} x^{7}) + \frac{d}{d x} [- 100 x] + \frac{d}{d x} [- 150 y] + \frac{d}{d x} [2021]$

Step 2.17

Combine $\frac{y^{3} (8 {(y^{3} x^{8})}^{- \frac{11}{12}})}{12}$ and $x^{7}$ .

$300 \frac{y^{3} (8 {(y^{3} x^{8})}^{- \frac{11}{12}}) x^{7}}{12} + \frac{d}{d x} [- 100 x] + \frac{d}{d x} [- 150 y] + \frac{d}{d x} [2021]$

Step 2.18

Move ${(y^{3} x^{8})}^{- \frac{11}{12}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$300 \frac{y^{3} \cdot 8 x^{7}}{12 {(y^{3} x^{8})}^{\frac{11}{12}}} + \frac{d}{d x} [- 100 x] + \frac{d}{d x} [- 150 y] + \frac{d}{d x} [2021]$

Step 2.19

Move $8$ to the left of $y^{3}$ .

$300 \frac{8 \cdot y^{3} x^{7}}{12 {(y^{3} x^{8})}^{\frac{11}{12}}} + \frac{d}{d x} [- 100 x] + \frac{d}{d x} [- 150 y] + \frac{d}{d x} [2021]$

Step 2.20

Factor $4$ out of $8 y^{3} x^{7}$ .

$300 \frac{4 (2 y^{3} x^{7})}{12 {(y^{3} x^{8})}^{\frac{11}{12}}} + \frac{d}{d x} [- 100 x] + \frac{d}{d x} [- 150 y] + \frac{d}{d x} [2021]$

Step 2.21

Cancel the common factors.

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Step 2.21.1

Factor $4$ out of $12 {(y^{3} x^{8})}^{\frac{11}{12}}$ .

$300 \frac{4 (2 y^{3} x^{7})}{4 (3 {(y^{3} x^{8})}^{\frac{11}{12}})} + \frac{d}{d x} [- 100 x] + \frac{d}{d x} [- 150 y] + \frac{d}{d x} [2021]$

Step 2.21.2

Cancel the common factor.

$300 \frac{4 (2 y^{3} x^{7})}{4 (3 {(y^{3} x^{8})}^{\frac{11}{12}})} + \frac{d}{d x} [- 100 x] + \frac{d}{d x} [- 150 y] + \frac{d}{d x} [2021]$

Step 2.21.3

Rewrite the expression.

$300 \frac{2 y^{3} x^{7}}{3 {(y^{3} x^{8})}^{\frac{11}{12}}} + \frac{d}{d x} [- 100 x] + \frac{d}{d x} [- 150 y] + \frac{d}{d x} [2021]$

Step 2.22

Combine $300$ and $\frac{2 y^{3} x^{7}}{3 {(y^{3} x^{8})}^{\frac{11}{12}}}$ .

$\frac{300 (2 y^{3} x^{7})}{3 {(y^{3} x^{8})}^{\frac{11}{12}}} + \frac{d}{d x} [- 100 x] + \frac{d}{d x} [- 150 y] + \frac{d}{d x} [2021]$

Step 2.23

Multiply $2$ by $300$ .

$\frac{600 (y^{3} x^{7})}{3 {(y^{3} x^{8})}^{\frac{11}{12}}} + \frac{d}{d x} [- 100 x] + \frac{d}{d x} [- 150 y] + \frac{d}{d x} [2021]$

Step 2.24

Factor $3$ out of $600 y^{3} x^{7}$ .

$\frac{3 (200 y^{3} x^{7})}{3 {(y^{3} x^{8})}^{\frac{11}{12}}} + \frac{d}{d x} [- 100 x] + \frac{d}{d x} [- 150 y] + \frac{d}{d x} [2021]$

Step 2.25

Cancel the common factors.

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Step 2.25.1

Factor $3$ out of $3 {(y^{3} x^{8})}^{\frac{11}{12}}$ .

$\frac{3 (200 y^{3} x^{7})}{3 ({(y^{3} x^{8})}^{\frac{11}{12}})} + \frac{d}{d x} [- 100 x] + \frac{d}{d x} [- 150 y] + \frac{d}{d x} [2021]$

Step 2.25.2

Cancel the common factor.

$\frac{3 (200 y^{3} x^{7})}{3 {(y^{3} x^{8})}^{\frac{11}{12}}} + \frac{d}{d x} [- 100 x] + \frac{d}{d x} [- 150 y] + \frac{d}{d x} [2021]$

Step 2.25.3

Rewrite the expression.

$\frac{200 y^{3} x^{7}}{{(y^{3} x^{8})}^{\frac{11}{12}}} + \frac{d}{d x} [- 100 x] + \frac{d}{d x} [- 150 y] + \frac{d}{d x} [2021]$

Step 3

Evaluate $\frac{d}{d x} [- 100 x]$ .

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Step 3.1

Since $- 100$ is constant with respect to $x$ , the derivative of $- 100 x$ with respect to $x$ is $- 100 \frac{d}{d x} [x]$ .

$\frac{200 y^{3} x^{7}}{{(y^{3} x^{8})}^{\frac{11}{12}}} - 100 \frac{d}{d x} [x] + \frac{d}{d x} [- 150 y] + \frac{d}{d x} [2021]$

Step 3.2

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

$\frac{200 y^{3} x^{7}}{{(y^{3} x^{8})}^{\frac{11}{12}}} - 100 \cdot 1 + \frac{d}{d x} [- 150 y] + \frac{d}{d x} [2021]$

Step 3.3

Multiply $- 100$ by $1$ .

$\frac{200 y^{3} x^{7}}{{(y^{3} x^{8})}^{\frac{11}{12}}} - 100 + \frac{d}{d x} [- 150 y] + \frac{d}{d x} [2021]$

Step 4

Differentiate using the Constant Rule.

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Step 4.1

Since $- 150 y$ is constant with respect to $x$ , the derivative of $- 150 y$ with respect to $x$ is $0$ .

$\frac{200 y^{3} x^{7}}{{(y^{3} x^{8})}^{\frac{11}{12}}} - 100 + 0 + \frac{d}{d x} [2021]$

Step 4.2

Since $2021$ is constant with respect to $x$ , the derivative of $2021$ with respect to $x$ is $0$ .

$\frac{200 y^{3} x^{7}}{{(y^{3} x^{8})}^{\frac{11}{12}}} - 100 + 0 + 0$

Step 5

Simplify.

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Step 5.1

Apply the product rule to $y^{3} x^{8}$ .

$\frac{200 y^{3} x^{7}}{{(y^{3})}^{\frac{11}{12}} {(x^{8})}^{\frac{11}{12}}} - 100 + 0 + 0$

Step 5.2

Combine terms.

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Step 5.2.1

Multiply the exponents in ${(y^{3})}^{\frac{11}{12}}$ .

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Step 5.2.1.1

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

$\frac{200 y^{3} x^{7}}{y^{3 (\frac{11}{12})} {(x^{8})}^{\frac{11}{12}}} - 100 + 0 + 0$

Step 5.2.1.2

Cancel the common factor of $3$ .

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Step 5.2.1.2.1

Factor $3$ out of $12$ .

$\frac{200 y^{3} x^{7}}{y^{3 \frac{11}{3 (4)}} {(x^{8})}^{\frac{11}{12}}} - 100 + 0 + 0$

Step 5.2.1.2.2

Cancel the common factor.

$\frac{200 y^{3} x^{7}}{y^{3 \frac{11}{3 \cdot 4}} {(x^{8})}^{\frac{11}{12}}} - 100 + 0 + 0$

Step 5.2.1.2.3

Rewrite the expression.

$\frac{200 y^{3} x^{7}}{y^{\frac{11}{4}} {(x^{8})}^{\frac{11}{12}}} - 100 + 0 + 0$

Step 5.2.2

Multiply the exponents in ${(x^{8})}^{\frac{11}{12}}$ .

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Step 5.2.2.1

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

$\frac{200 y^{3} x^{7}}{y^{\frac{11}{4}} x^{8 (\frac{11}{12})}} - 100 + 0 + 0$

Step 5.2.2.2

Cancel the common factor of $4$ .

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Step 5.2.2.2.1

Factor $4$ out of $8$ .

$\frac{200 y^{3} x^{7}}{y^{\frac{11}{4}} x^{4 (2) \frac{11}{12}}} - 100 + 0 + 0$

Step 5.2.2.2.2

Factor $4$ out of $12$ .

$\frac{200 y^{3} x^{7}}{y^{\frac{11}{4}} x^{4 \cdot 2 \frac{11}{4 \cdot 3}}} - 100 + 0 + 0$

Step 5.2.2.2.3

Cancel the common factor.

$\frac{200 y^{3} x^{7}}{y^{\frac{11}{4}} x^{4 \cdot 2 \frac{11}{4 \cdot 3}}} - 100 + 0 + 0$

Step 5.2.2.2.4

Rewrite the expression.

$\frac{200 y^{3} x^{7}}{y^{\frac{11}{4}} x^{2 (\frac{11}{3})}} - 100 + 0 + 0$

Step 5.2.2.3

Combine $2$ and $\frac{11}{3}$ .

$\frac{200 y^{3} x^{7}}{y^{\frac{11}{4}} x^{\frac{2 \cdot 11}{3}}} - 100 + 0 + 0$

Step 5.2.2.4

Multiply $2$ by $11$ .

$\frac{200 y^{3} x^{7}}{y^{\frac{11}{4}} x^{\frac{22}{3}}} - 100 + 0 + 0$

Step 5.2.3

Move $x^{7}$ to the denominator using the negative exponent rule $b^{n} = \frac{1}{b^{- n}}$ .

$\frac{200 y^{3}}{y^{\frac{11}{4}} x^{\frac{22}{3}} x^{- 7}} - 100 + 0 + 0$

Step 5.2.4

Multiply $x^{\frac{22}{3}}$ by $x^{- 7}$ by adding the exponents.

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Step 5.2.4.1

Move $x^{- 7}$ .

$\frac{200 y^{3}}{y^{\frac{11}{4}} (x^{- 7} x^{\frac{22}{3}})} - 100 + 0 + 0$

Step 5.2.4.2

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

$\frac{200 y^{3}}{y^{\frac{11}{4}} x^{- 7 + \frac{22}{3}}} - 100 + 0 + 0$

Step 5.2.4.3

To write $- 7$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{200 y^{3}}{y^{\frac{11}{4}} x^{- 7 \cdot \frac{3}{3} + \frac{22}{3}}} - 100 + 0 + 0$

Step 5.2.4.4

Combine $- 7$ and $\frac{3}{3}$ .

$\frac{200 y^{3}}{y^{\frac{11}{4}} x^{\frac{- 7 \cdot 3}{3} + \frac{22}{3}}} - 100 + 0 + 0$

Step 5.2.4.5

Combine the numerators over the common denominator.

$\frac{200 y^{3}}{y^{\frac{11}{4}} x^{\frac{- 7 \cdot 3 + 22}{3}}} - 100 + 0 + 0$

Step 5.2.4.6

Simplify the numerator.

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Step 5.2.4.6.1

Multiply $- 7$ by $3$ .

$\frac{200 y^{3}}{y^{\frac{11}{4}} x^{\frac{- 21 + 22}{3}}} - 100 + 0 + 0$

Step 5.2.4.6.2

Add $- 21$ and $22$ .

$\frac{200 y^{3}}{y^{\frac{11}{4}} x^{\frac{1}{3}}} - 100 + 0 + 0$

Step 5.2.5

Move $y^{\frac{11}{4}}$ to the numerator using the negative exponent rule $\frac{1}{b^{n}} = b^{- n}$ .

$\frac{200 y^{3} y^{- \frac{11}{4}}}{x^{\frac{1}{3}}} - 100 + 0 + 0$

Step 5.2.6

Multiply $y^{3}$ by $y^{- \frac{11}{4}}$ by adding the exponents.

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Step 5.2.6.1

Move $y^{- \frac{11}{4}}$ .

$\frac{200 (y^{- \frac{11}{4}} y^{3})}{x^{\frac{1}{3}}} - 100 + 0 + 0$

Step 5.2.6.2

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

$\frac{200 y^{- \frac{11}{4} + 3}}{x^{\frac{1}{3}}} - 100 + 0 + 0$

Step 5.2.6.3

To write $3$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$\frac{200 y^{- \frac{11}{4} + 3 \cdot \frac{4}{4}}}{x^{\frac{1}{3}}} - 100 + 0 + 0$

Step 5.2.6.4

Combine $3$ and $\frac{4}{4}$ .

$\frac{200 y^{- \frac{11}{4} + \frac{3 \cdot 4}{4}}}{x^{\frac{1}{3}}} - 100 + 0 + 0$

Step 5.2.6.5

Combine the numerators over the common denominator.

$\frac{200 y^{\frac{- 11 + 3 \cdot 4}{4}}}{x^{\frac{1}{3}}} - 100 + 0 + 0$

Step 5.2.6.6

Simplify the numerator.

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Step 5.2.6.6.1

Multiply $3$ by $4$ .

$\frac{200 y^{\frac{- 11 + 12}{4}}}{x^{\frac{1}{3}}} - 100 + 0 + 0$

Step 5.2.6.6.2

Add $- 11$ and $12$ .

$\frac{200 y^{\frac{1}{4}}}{x^{\frac{1}{3}}} - 100 + 0 + 0$

Step 5.2.7

Add $\frac{200 y^{\frac{1}{4}}}{x^{\frac{1}{3}}} - 100$ and $0$ .

$\frac{200 y^{\frac{1}{4}}}{x^{\frac{1}{3}}} - 100 + 0$

Step 5.2.8

Add $\frac{200 y^{\frac{1}{4}}}{x^{\frac{1}{3}}} - 100$ and $0$ .

$\frac{200 y^{\frac{1}{4}}}{x^{\frac{1}{3}}} - 100$