Calculus Examples

${(9 - 5 x)}^{2}$

Step 1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{2}$ and $g (x) = 9 - 5 x$ .

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Step 1.1

To apply the Chain Rule, set $u$ as $9 - 5 x$ .

$\frac{d}{d u} [u^{2}] \frac{d}{d x} [9 - 5 x]$

Step 1.2

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

$2 u \frac{d}{d x} [9 - 5 x]$

Step 1.3

Replace all occurrences of $u$ with $9 - 5 x$ .

$2 (9 - 5 x) \frac{d}{d x} [9 - 5 x]$

Step 2

Differentiate.

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Step 2.1

By the Sum Rule, the derivative of $9 - 5 x$ with respect to $x$ is $\frac{d}{d x} [9] + \frac{d}{d x} [- 5 x]$ .

$2 (9 - 5 x) (\frac{d}{d x} [9] + \frac{d}{d x} [- 5 x])$

Step 2.2

Since $9$ is constant with respect to $x$ , the derivative of $9$ with respect to $x$ is $0$ .

$2 (9 - 5 x) (0 + \frac{d}{d x} [- 5 x])$

Step 3

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

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Step 3.1

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

$2 (9 - 5 x) (0 - 5 \frac{d}{d x} [x])$

Step 3.2

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

$2 (9 - 5 x) (0 - 5 \cdot 1)$

Step 3.3

Multiply $- 5$ by $1$ .

$2 (9 - 5 x) (0 - 5)$

Step 4

Subtract $5$ from $0$ .

$2 (9 - 5 x) \cdot - 5$

Step 5

Simplify.

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Step 5.1

Apply the distributive property.

$(2 \cdot 9 + 2 (- 5 x)) \cdot - 5$

Step 5.2

Apply the distributive property.

$2 \cdot 9 \cdot - 5 + 2 (- 5 x) \cdot - 5$

Step 5.3

Combine terms.

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Step 5.3.1

Multiply $2$ by $9$ .

$18 \cdot - 5 + 2 (- 5 x) \cdot - 5$

Step 5.3.2

Multiply $18$ by $- 5$ .

$- 90 + 2 (- 5 x) \cdot - 5$

Step 5.3.3

Multiply $- 5$ by $2$ .

$- 90 - 10 x \cdot - 5$

Step 5.3.4

Multiply $- 5$ by $- 10$ .

$- 90 + 50 x$

Step 5.4

Reorder terms.

$50 x - 90$