Calculus Examples

$y = - 7 x^{- 5} + 4 x^{2} - 19 x + 2$

Step 1

By the Sum Rule, the derivative of $- 7 x^{- 5} + 4 x^{2} - 19 x + 2$ with respect to $x$ is $\frac{d}{d x} [- 7 x^{- 5}] + \frac{d}{d x} [4 x^{2}] + \frac{d}{d x} [- 19 x] + \frac{d}{d x} [2]$ .

$\frac{d}{d x} [- 7 x^{- 5}] + \frac{d}{d x} [4 x^{2}] + \frac{d}{d x} [- 19 x] + \frac{d}{d x} [2]$

Step 2

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

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Step 2.1

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

$- 7 \frac{d}{d x} [x^{- 5}] + \frac{d}{d x} [4 x^{2}] + \frac{d}{d x} [- 19 x] + \frac{d}{d x} [2]$

Step 2.2

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

$- 7 (- 5 x^{- 6}) + \frac{d}{d x} [4 x^{2}] + \frac{d}{d x} [- 19 x] + \frac{d}{d x} [2]$

Step 2.3

Multiply $- 5$ by $- 7$ .

$35 x^{- 6} + \frac{d}{d x} [4 x^{2}] + \frac{d}{d x} [- 19 x] + \frac{d}{d x} [2]$

Step 3

Evaluate $\frac{d}{d x} [4 x^{2}]$ .

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Step 3.1

Since $4$ is constant with respect to $x$ , the derivative of $4 x^{2}$ with respect to $x$ is $4 \frac{d}{d x} [x^{2}]$ .

$35 x^{- 6} + 4 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 19 x] + \frac{d}{d x} [2]$

Step 3.2

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

$35 x^{- 6} + 4 (2 x) + \frac{d}{d x} [- 19 x] + \frac{d}{d x} [2]$

Step 3.3

Multiply $2$ by $4$ .

$35 x^{- 6} + 8 x + \frac{d}{d x} [- 19 x] + \frac{d}{d x} [2]$

Step 4

Evaluate $\frac{d}{d x} [- 19 x]$ .

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Step 4.1

Since $- 19$ is constant with respect to $x$ , the derivative of $- 19 x$ with respect to $x$ is $- 19 \frac{d}{d x} [x]$ .

$35 x^{- 6} + 8 x - 19 \frac{d}{d x} [x] + \frac{d}{d x} [2]$

Step 4.2

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

$35 x^{- 6} + 8 x - 19 \cdot 1 + \frac{d}{d x} [2]$

Step 4.3

Multiply $- 19$ by $1$ .

$35 x^{- 6} + 8 x - 19 + \frac{d}{d x} [2]$

Step 5

Since $2$ is constant with respect to $x$ , the derivative of $2$ with respect to $x$ is $0$ .

$35 x^{- 6} + 8 x - 19 + 0$

Step 6

Simplify.

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Step 6.1

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

$35 \frac{1}{x^{6}} + 8 x - 19 + 0$

Step 6.2

Combine terms.

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Step 6.2.1

Combine $35$ and $\frac{1}{x^{6}}$ .

$\frac{35}{x^{6}} + 8 x - 19 + 0$

Step 6.2.2

Add $\frac{35}{x^{6}} + 8 x - 19$ and $0$ .

$\frac{35}{x^{6}} + 8 x - 19$

Step 6.3

Reorder terms.

$8 x + \frac{35}{x^{6}} - 19$