Calculus Examples

$y = 2 x^{2} - 5 x + 3 x^{- 2} + 4 x^{- 3}$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (y) = \frac{d}{d x} (2 x^{2} - 5 x + 3 x^{- 2} + 4 x^{- 3})$

Step 2

The derivative of $y$ with respect to $x$ is $y'$ .

$y'$

Step 3

Differentiate the right side of the equation.

Tap for more steps...

Step 3.1

By the Sum Rule, the derivative of $2 x^{2} - 5 x + 3 x^{- 2} + 4 x^{- 3}$ with respect to $x$ is $\frac{d}{d x} [2 x^{2}] + \frac{d}{d x} [- 5 x] + \frac{d}{d x} [3 x^{- 2}] + \frac{d}{d x} [4 x^{- 3}]$ .

$\frac{d}{d x} [2 x^{2}] + \frac{d}{d x} [- 5 x] + \frac{d}{d x} [3 x^{- 2}] + \frac{d}{d x} [4 x^{- 3}]$

Step 3.2

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

Tap for more steps...

Step 3.2.1

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

$2 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 5 x] + \frac{d}{d x} [3 x^{- 2}] + \frac{d}{d x} [4 x^{- 3}]$

Step 3.2.2

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

$2 (2 x) + \frac{d}{d x} [- 5 x] + \frac{d}{d x} [3 x^{- 2}] + \frac{d}{d x} [4 x^{- 3}]$

Step 3.2.3

Multiply $2$ by $2$ .

$4 x + \frac{d}{d x} [- 5 x] + \frac{d}{d x} [3 x^{- 2}] + \frac{d}{d x} [4 x^{- 3}]$

Step 3.3

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

Tap for more steps...

Step 3.3.1

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

$4 x - 5 \frac{d}{d x} [x] + \frac{d}{d x} [3 x^{- 2}] + \frac{d}{d x} [4 x^{- 3}]$

Step 3.3.2

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

$4 x - 5 \cdot 1 + \frac{d}{d x} [3 x^{- 2}] + \frac{d}{d x} [4 x^{- 3}]$

Step 3.3.3

Multiply $- 5$ by $1$ .

$4 x - 5 + \frac{d}{d x} [3 x^{- 2}] + \frac{d}{d x} [4 x^{- 3}]$

Step 3.4

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

Tap for more steps...

Step 3.4.1

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

$4 x - 5 + 3 \frac{d}{d x} [x^{- 2}] + \frac{d}{d x} [4 x^{- 3}]$

Step 3.4.2

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

$4 x - 5 + 3 (- 2 x^{- 3}) + \frac{d}{d x} [4 x^{- 3}]$

Step 3.4.3

Multiply $- 2$ by $3$ .

$4 x - 5 - 6 x^{- 3} + \frac{d}{d x} [4 x^{- 3}]$

Step 3.5

Evaluate $\frac{d}{d x} [4 x^{- 3}]$ .

Tap for more steps...

Step 3.5.1

Since $4$ is constant with respect to $x$ , the derivative of $4 x^{- 3}$ with respect to $x$ is $4 \frac{d}{d x} [x^{- 3}]$ .

$4 x - 5 - 6 x^{- 3} + 4 \frac{d}{d x} [x^{- 3}]$

Step 3.5.2

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

$4 x - 5 - 6 x^{- 3} + 4 (- 3 x^{- 4})$

Step 3.5.3

Multiply $- 3$ by $4$ .

$4 x - 5 - 6 x^{- 3} - 12 x^{- 4}$

Step 3.6

Simplify.

Tap for more steps...

Step 3.6.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$4 x - 5 - 6 \frac{1}{x^{3}} - 12 x^{- 4}$

Step 3.6.2

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$4 x - 5 - 6 \frac{1}{x^{3}} - 12 \frac{1}{x^{4}}$

Step 3.6.3

Combine terms.

Tap for more steps...

Step 3.6.3.1

Combine $- 6$ and $\frac{1}{x^{3}}$ .

$4 x - 5 + \frac{- 6}{x^{3}} - 12 \frac{1}{x^{4}}$

Step 3.6.3.2

Move the negative in front of the fraction.

$4 x - 5 - \frac{6}{x^{3}} - 12 \frac{1}{x^{4}}$

Step 3.6.3.3

Combine $- 12$ and $\frac{1}{x^{4}}$ .

$4 x - 5 - \frac{6}{x^{3}} + \frac{- 12}{x^{4}}$

Step 3.6.3.4

Move the negative in front of the fraction.

$4 x - 5 - \frac{6}{x^{3}} - \frac{12}{x^{4}}$

Step 3.6.4

Reorder terms.

$4 x - \frac{6}{x^{3}} - \frac{12}{x^{4}} - 5$

Step 4

Reform the equation by setting the left side equal to the right side.

$y' = 4 x - \frac{6}{x^{3}} - \frac{12}{x^{4}} - 5$

Step 5

Replace $y'$ with $\frac{d y}{d x}$ .

$\frac{d y}{d x} = 4 x - \frac{6}{x^{3}} - \frac{12}{x^{4}} - 5$