Calculus Examples

$(1 + 7 x^{2}) (x - x^{2})$

Step 1

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = 1 + 7 x^{2}$ and $g (x) = x - x^{2}$ .

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

Step 2

Differentiate.

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

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

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

Step 2.5

Multiply $2$ by $- 1$ .

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

Step 2.6

By the Sum Rule, the derivative of $1 + 7 x^{2}$ with respect to $x$ is $\frac{d}{d x} [1] + \frac{d}{d x} [7 x^{2}]$ .

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

Step 2.7

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$(1 + 7 x^{2}) (1 - 2 x) + (x - x^{2}) (0 + \frac{d}{d x} [7 x^{2}])$

Step 2.8

Add $0$ and $\frac{d}{d x} [7 x^{2}]$ .

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

Step 2.9

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

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

Step 2.10

Move $7$ to the left of $x - x^{2}$ .

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

Step 2.11

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

$(1 + 7 x^{2}) (1 - 2 x) + 7 (x - x^{2}) (2 x)$

Step 2.12

Multiply $2$ by $7$ .

$(1 + 7 x^{2}) (1 - 2 x) + 14 (x - x^{2}) x$

Step 3

Simplify.

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Step 3.1