Calculus Examples

$\frac{3 x^{2}}{2} - \frac{7}{5 x^{2}}$

Step 1

By the Sum Rule, the derivative of $\frac{3 x^{2}}{2} - \frac{7}{5 x^{2}}$ with respect to $x$ is $\frac{d}{d x} [\frac{3 x^{2}}{2}] + \frac{d}{d x} [- \frac{7}{5 x^{2}}]$ .

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

Step 2

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

Tap for more steps...

Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

Multiply $2$ by $3$ .

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

Step 2.5

Combine $\frac{6}{2}$ and $x$ .

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

Step 2.6

Cancel the common factor of $6$ and $2$ .

Tap for more steps...

Step 2.6.1

Factor $2$ out of $6 x$ .

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

Step 2.6.2

Cancel the common factors.

Tap for more steps...

Step 2.6.2.1

Factor $2$ out of $2$ .

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

Step 2.6.2.2

Cancel the common factor.

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

Step 2.6.2.3

Rewrite the expression.

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

Step 2.6.2.4

Divide $3 x$ by $1$ .

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

Step 3

Evaluate $\frac{d}{d x} [- \frac{7}{5 x^{2}}]$ .

Tap for more steps...

Step 3.1

Since $- \frac{7}{5}$ is constant with respect to $x$ , the derivative of $- \frac{7}{5 x^{2}}$ with respect to $x$ is $- \frac{7}{5} \frac{d}{d x} [\frac{1}{x^{2}}]$ .

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

Step 3.2

Rewrite $\frac{1}{x^{2}}$ as ${(x^{2})}^{- 1}$ .

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

Step 3.3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{- 1}$ and $g (x) = x^{2}$ .

Tap for more steps...

Step 3.3.1

To apply the Chain Rule, set $u$ as $x^{2}$ .

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

Step 3.3.2

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

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

Step 3.3.3

Replace all occurrences of $u$ with $x^{2}$ .

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

Step 3.4

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

$3 x - \frac{7}{5} (- {(x^{2})}^{- 2} (2 x))$

Step 3.5

Multiply the exponents in ${(x^{2})}^{- 2}$ .

Tap for more steps...

Step 3.5.1

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

$3 x - \frac{7}{5} (- x^{2 \cdot - 2} (2 x))$

Step 3.5.2

Multiply $2$ by $- 2$ .

$3 x - \frac{7}{5} (- x^{- 4} (2 x))$

Step 3.6

Multiply $2$ by $- 1$ .

$3 x - \frac{7}{5} (- 2 x^{- 4} x)$

Step 3.7

Raise $x$ to the power of $1$ .

$3 x - \frac{7}{5} (- 2 (x^{1} x^{- 4}))$

Step 3.8

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

$3 x - \frac{7}{5} (- 2 x^{1 - 4})$

Step 3.9

Subtract $4$ from $1$ .

$3 x - \frac{7}{5} (- 2 x^{- 3})$

Step 3.10

Multiply $- 2$ by $- 1$ .

$3 x + 2 (\frac{7}{5}) x^{- 3}$

Step 3.11

Combine $2$ and $\frac{7}{5}$ .

$3 x + \frac{2 \cdot 7}{5} x^{- 3}$

Step 3.12

Multiply $2$ by $7$ .

$3 x + \frac{14}{5} x^{- 3}$

Step 3.13

Combine $\frac{14}{5}$ and $x^{- 3}$ .

$3 x + \frac{14 x^{- 3}}{5}$

Step 3.14

Move $x^{- 3}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$3 x + \frac{14}{5 x^{3}}$