Calculus Examples

$6 y^{\frac{5}{2}} - 5 x y - 7 x$

Step 1

Find the first derivative.

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Step 1.1

By the Sum Rule, the derivative of $6 y^{\frac{5}{2}} - 5 x y - 7 x$ with respect to $y$ is $\frac{d}{d y} [6 y^{\frac{5}{2}}] + \frac{d}{d y} [- 5 x y] + \frac{d}{d y} [- 7 x]$ .

$\frac{d}{d y} [6 y^{\frac{5}{2}}] + \frac{d}{d y} [- 5 x y] + \frac{d}{d y} [- 7 x]$

Step 1.2

Evaluate $\frac{d}{d y} [6 y^{\frac{5}{2}}]$ .

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Step 1.2.1

Since $6$ is constant with respect to $y$ , the derivative of $6 y^{\frac{5}{2}}$ with respect to $y$ is $6 \frac{d}{d y} [y^{\frac{5}{2}}]$ .

$6 \frac{d}{d y} [y^{\frac{5}{2}}] + \frac{d}{d y} [- 5 x y] + \frac{d}{d y} [- 7 x]$

Step 1.2.2

Differentiate using the Power Rule which states that $\frac{d}{d y} [y^{n}]$ is $n y^{n - 1}$ where $n = \frac{5}{2}$ .

$6 (\frac{5}{2} y^{\frac{5}{2} - 1}) + \frac{d}{d y} [- 5 x y] + \frac{d}{d y} [- 7 x]$

Step 1.2.3

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

$6 (\frac{5}{2} y^{\frac{5}{2} - 1 \cdot \frac{2}{2}}) + \frac{d}{d y} [- 5 x y] + \frac{d}{d y} [- 7 x]$

Step 1.2.4

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

$6 (\frac{5}{2} y^{\frac{5}{2} + \frac{- 1 \cdot 2}{2}}) + \frac{d}{d y} [- 5 x y] + \frac{d}{d y} [- 7 x]$

Step 1.2.5

Combine the numerators over the common denominator.

$6 (\frac{5}{2} y^{\frac{5 - 1 \cdot 2}{2}}) + \frac{d}{d y} [- 5 x y] + \frac{d}{d y} [- 7 x]$

Step 1.2.6

Simplify the numerator.

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Step 1.2.6.1

Multiply $- 1$ by $2$ .

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

Step 1.2.6.2

Subtract $2$ from $5$ .

$6 (\frac{5}{2} y^{\frac{3}{2}}) + \frac{d}{d y} [- 5 x y] + \frac{d}{d y} [- 7 x]$

Step 1.2.7

Combine $\frac{5}{2}$ and $y^{\frac{3}{2}}$ .

$6 \frac{5 y^{\frac{3}{2}}}{2} + \frac{d}{d y} [- 5 x y] + \frac{d}{d y} [- 7 x]$

Step 1.2.8

Combine $6$ and $\frac{5 y^{\frac{3}{2}}}{2}$ .

$\frac{6 (5 y^{\frac{3}{2}})}{2} + \frac{d}{d y} [- 5 x y] + \frac{d}{d y} [- 7 x]$

Step 1.2.9

Multiply $5$ by $6$ .

$\frac{30 y^{\frac{3}{2}}}{2} + \frac{d}{d y} [- 5 x y] + \frac{d}{d y} [- 7 x]$

Step 1.2.10

Factor $2$ out of $30 y^{\frac{3}{2}}$ .

$\frac{2 (15 y^{\frac{3}{2}})}{2} + \frac{d}{d y} [- 5 x y] + \frac{d}{d y} [- 7 x]$

Step 1.2.11

Cancel the common factors.

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Step 1.2.11.1

Factor $2$ out of $2$ .

$\frac{2 (15 y^{\frac{3}{2}})}{2 (1)} + \frac{d}{d y} [- 5 x y] + \frac{d}{d y} [- 7 x]$

Step 1.2.11.2

Cancel the common factor.

$\frac{2 (15 y^{\frac{3}{2}})}{2 \cdot 1} + \frac{d}{d y} [- 5 x y] + \frac{d}{d y} [- 7 x]$

Step 1.2.11.3

Rewrite the expression.

$\frac{15 y^{\frac{3}{2}}}{1} + \frac{d}{d y} [- 5 x y] + \frac{d}{d y} [- 7 x]$

Step 1.2.11.4

Divide $15 y^{\frac{3}{2}}$ by $1$ .

$15 y^{\frac{3}{2}} + \frac{d}{d y} [- 5 x y] + \frac{d}{d y} [- 7 x]$

Step 1.3

Evaluate $\frac{d}{d y} [- 5 x y]$ .

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Step 1.3.1

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

$15 y^{\frac{3}{2}} - 5 x \frac{d}{d y} [y] + \frac{d}{d y} [- 7 x]$

Step 1.3.2

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

$15 y^{\frac{3}{2}} - 5 x \cdot 1 + \frac{d}{d y} [- 7 x]$

Step 1.3.3

Multiply $- 5$ by $1$ .

$15 y^{\frac{3}{2}} - 5 x + \frac{d}{d y} [- 7 x]$

Step 1.4

Since $- 7 x$ is constant with respect to $y$ , the derivative of $- 7 x$ with respect to $y$ is $0$ .

$15 y^{\frac{3}{2}} - 5 x + 0$

Step 1.5

Simplify.

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Step 1.5.1

Add $15 y^{\frac{3}{2}} - 5 x$ and $0$ .

$15 y^{\frac{3}{2}} - 5 x$

Step 1.5.2

Reorder terms.

$f' (y) = - 5 x + 15 y^{\frac{3}{2}}$

Step 2

Find the second derivative.

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Step 2.1

Differentiate.

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Step 2.1.1

By the Sum Rule, the derivative of $- 5 x + 15 y^{\frac{3}{2}}$ with respect to $y$ is $\frac{d}{d y} [- 5 x] + \frac{d}{d y} [15 y^{\frac{3}{2}}]$ .

$\frac{d}{d y} [- 5 x] + \frac{d}{d y} [15 y^{\frac{3}{2}}]$

Step 2.1.2

Since $- 5 x$ is constant with respect to $y$ , the derivative of $- 5 x$ with respect to $y$ is $0$ .

$0 + \frac{d}{d y} [15 y^{\frac{3}{2}}]$

Step 2.2

Evaluate $\frac{d}{d y} [15 y^{\frac{3}{2}}]$ .

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Step 2.2.1

Since $15$ is constant with respect to $y$ , the derivative of $15 y^{\frac{3}{2}}$ with respect to $y$ is $15 \frac{d}{d y} [y^{\frac{3}{2}}]$ .

$0 + 15 \frac{d}{d y} [y^{\frac{3}{2}}]$

Step 2.2.2

Differentiate using the Power Rule which states that $\frac{d}{d y} [y^{n}]$ is $n y^{n - 1}$ where $n = \frac{3}{2}$ .

$0 + 15 (\frac{3}{2} y^{\frac{3}{2} - 1})$

Step 2.2.3

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

$0 + 15 (\frac{3}{2} y^{\frac{3}{2} - 1 \cdot \frac{2}{2}})$

Step 2.2.4

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

$0 + 15 (\frac{3}{2} y^{\frac{3}{2} + \frac{- 1 \cdot 2}{2}})$

Step 2.2.5

Combine the numerators over the common denominator.

$0 + 15 (\frac{3}{2} y^{\frac{3 - 1 \cdot 2}{2}})$

Step 2.2.6

Simplify the numerator.

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Step 2.2.6.1

Multiply $- 1$ by $2$ .

$0 + 15 (\frac{3}{2} y^{\frac{3 - 2}{2}})$

Step 2.2.6.2

Subtract $2$ from $3$ .

$0 + 15 (\frac{3}{2} y^{\frac{1}{2}})$

Step 2.2.7

Combine $\frac{3}{2}$ and $y^{\frac{1}{2}}$ .

$0 + 15 \frac{3 y^{\frac{1}{2}}}{2}$

Step 2.2.8

Combine $15$ and $\frac{3 y^{\frac{1}{2}}}{2}$ .

$0 + \frac{15 (3 y^{\frac{1}{2}})}{2}$

Step 2.2.9

Multiply $3$ by $15$ .

$0 + \frac{45 y^{\frac{1}{2}}}{2}$

Step 2.3

Add $0$ and $\frac{45 y^{\frac{1}{2}}}{2}$ .

$f'' (y) = \frac{45 y^{\frac{1}{2}}}{2}$