Calculus Examples

$10 x^{9} y^{5} + 7 x^{4} y^{8}$

Step 1

Find the first derivative.

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Step 1.1

By the Sum Rule, the derivative of $10 x^{9} y^{5} + 7 x^{4} y^{8}$ with respect to $x$ is $\frac{d}{d x} [10 x^{9} y^{5}] + \frac{d}{d x} [7 x^{4} y^{8}]$ .

$\frac{d}{d x} [10 x^{9} y^{5}] + \frac{d}{d x} [7 x^{4} y^{8}]$

Step 1.2

Evaluate $\frac{d}{d x} [10 x^{9} y^{5}]$ .

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Step 1.2.1

Since $10 y^{5}$ is constant with respect to $x$ , the derivative of $10 x^{9} y^{5}$ with respect to $x$ is $10 y^{5} \frac{d}{d x} [x^{9}]$ .

$10 y^{5} \frac{d}{d x} [x^{9}] + \frac{d}{d x} [7 x^{4} y^{8}]$

Step 1.2.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 9$ .

$10 y^{5} (9 x^{8}) + \frac{d}{d x} [7 x^{4} y^{8}]$

Step 1.2.3

Multiply $9$ by $10$ .

$90 y^{5} x^{8} + \frac{d}{d x} [7 x^{4} y^{8}]$

Step 1.3

Evaluate $\frac{d}{d x} [7 x^{4} y^{8}]$ .

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Step 1.3.1

Since $7 y^{8}$ is constant with respect to $x$ , the derivative of $7 x^{4} y^{8}$ with respect to $x$ is $7 y^{8} \frac{d}{d x} [x^{4}]$ .

$90 y^{5} x^{8} + 7 y^{8} \frac{d}{d x} [x^{4}]$

Step 1.3.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 4$ .

$90 y^{5} x^{8} + 7 y^{8} (4 x^{3})$

Step 1.3.3

Multiply $4$ by $7$ .

$90 y^{5} x^{8} + 28 y^{8} x^{3}$

Step 1.4

Reorder terms.

$f' (x) = 90 x^{8} y^{5} + 28 y^{8} x^{3}$

Step 2

Find the second derivative.

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Step 2.1

By the Sum Rule, the derivative of $90 x^{8} y^{5} + 28 y^{8} x^{3}$ with respect to $x$ is $\frac{d}{d x} [90 x^{8} y^{5}] + \frac{d}{d x} [28 y^{8} x^{3}]$ .

$\frac{d}{d x} [90 x^{8} y^{5}] + \frac{d}{d x} [28 y^{8} x^{3}]$

Step 2.2

Evaluate $\frac{d}{d x} [90 x^{8} y^{5}]$ .

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Step 2.2.1

Since $90 y^{5}$ is constant with respect to $x$ , the derivative of $90 x^{8} y^{5}$ with respect to $x$ is $90 y^{5} \frac{d}{d x} [x^{8}]$ .

$90 y^{5} \frac{d}{d x} [x^{8}] + \frac{d}{d x} [28 y^{8} x^{3}]$

Step 2.2.2

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

$90 y^{5} (8 x^{7}) + \frac{d}{d x} [28 y^{8} x^{3}]$

Step 2.2.3

Multiply $8$ by $90$ .

$720 y^{5} x^{7} + \frac{d}{d x} [28 y^{8} x^{3}]$

Step 2.3

Evaluate $\frac{d}{d x} [28 y^{8} x^{3}]$ .

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Step 2.3.1

Since $28 y^{8}$ is constant with respect to $x$ , the derivative of $28 y^{8} x^{3}$ with respect to $x$ is $28 y^{8} \frac{d}{d x} [x^{3}]$ .

$720 y^{5} x^{7} + 28 y^{8} \frac{d}{d x} [x^{3}]$

Step 2.3.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

$720 y^{5} x^{7} + 28 y^{8} (3 x^{2})$

Step 2.3.3

Multiply $3$ by $28$ .

$720 y^{5} x^{7} + 84 y^{8} x^{2}$

Step 2.4

Reorder terms.

$f'' (x) = 720 x^{7} y^{5} + 84 y^{8} x^{2}$

Step 3

Find the third derivative.

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Step 3.1

By the Sum Rule, the derivative of $720 x^{7} y^{5} + 84 y^{8} x^{2}$ with respect to $x$ is $\frac{d}{d x} [720 x^{7} y^{5}] + \frac{d}{d x} [84 y^{8} x^{2}]$ .

$\frac{d}{d x} [720 x^{7} y^{5}] + \frac{d}{d x} [84 y^{8} x^{2}]$

Step 3.2

Evaluate $\frac{d}{d x} [720 x^{7} y^{5}]$ .

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Step 3.2.1

Since $720 y^{5}$ is constant with respect to $x$ , the derivative of $720 x^{7} y^{5}$ with respect to $x$ is $720 y^{5} \frac{d}{d x} [x^{7}]$ .

$720 y^{5} \frac{d}{d x} [x^{7}] + \frac{d}{d x} [84 y^{8} x^{2}]$

Step 3.2.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 7$ .

$720 y^{5} (7 x^{6}) + \frac{d}{d x} [84 y^{8} x^{2}]$

Step 3.2.3

Multiply $7$ by $720$ .

$5040 y^{5} x^{6} + \frac{d}{d x} [84 y^{8} x^{2}]$

Step 3.3

Evaluate $\frac{d}{d x} [84 y^{8} x^{2}]$ .

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Step 3.3.1

Since $84 y^{8}$ is constant with respect to $x$ , the derivative of $84 y^{8} x^{2}$ with respect to $x$ is $84 y^{8} \frac{d}{d x} [x^{2}]$ .

$5040 y^{5} x^{6} + 84 y^{8} \frac{d}{d x} [x^{2}]$

Step 3.3.2

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

$5040 y^{5} x^{6} + 84 y^{8} (2 x)$

Step 3.3.3

Multiply $2$ by $84$ .

$5040 y^{5} x^{6} + 168 y^{8} x$

Step 3.4

Reorder terms.

$f''' (x) = 5040 x^{6} y^{5} + 168 y^{8} x$

Step 4

Find the fourth derivative.

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Step 4.1

By the Sum Rule, the derivative of $5040 x^{6} y^{5} + 168 y^{8} x$ with respect to $x$ is $\frac{d}{d x} [5040 x^{6} y^{5}] + \frac{d}{d x} [168 y^{8} x]$ .

$\frac{d}{d x} [5040 x^{6} y^{5}] + \frac{d}{d x} [168 y^{8} x]$

Step 4.2

Evaluate $\frac{d}{d x} [5040 x^{6} y^{5}]$ .

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Step 4.2.1

Since $5040 y^{5}$ is constant with respect to $x$ , the derivative of $5040 x^{6} y^{5}$ with respect to $x$ is $5040 y^{5} \frac{d}{d x} [x^{6}]$ .

$5040 y^{5} \frac{d}{d x} [x^{6}] + \frac{d}{d x} [168 y^{8} x]$

Step 4.2.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 6$ .

$5040 y^{5} (6 x^{5}) + \frac{d}{d x} [168 y^{8} x]$

Step 4.2.3

Multiply $6$ by $5040$ .

$30240 y^{5} x^{5} + \frac{d}{d x} [168 y^{8} x]$

Step 4.3

Evaluate $\frac{d}{d x} [168 y^{8} x]$ .

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Step 4.3.1

Since $168 y^{8}$ is constant with respect to $x$ , the derivative of $168 y^{8} x$ with respect to $x$ is $168 y^{8} \frac{d}{d x} [x]$ .

$30240 y^{5} x^{5} + 168 y^{8} \frac{d}{d x} [x]$

Step 4.3.2

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

$30240 y^{5} x^{5} + 168 y^{8} \cdot 1$

Step 4.3.3

Multiply $168$ by $1$ .

$30240 y^{5} x^{5} + 168 y^{8}$

Step 4.4

Reorder terms.

$f^{4} (x) = 168 y^{8} + 30240 x^{5} y^{5}$