Calculus Examples

$h (x) = {(8 - 5 x)}^{3} - 7 {(8 - 5 x)}^{2} + 3 (8 - 5 x) - 1$

Step 1

Rewrite ${(8 - 5 x)}^{2}$ as $(8 - 5 x) (8 - 5 x)$ .

$\frac{d}{d x} [{(8 - 5 x)}^{3} - 7 ((8 - 5 x) (8 - 5 x)) + 3 (8 - 5 x) - 1]$

Step 2

Expand $(8 - 5 x) (8 - 5 x)$ using the FOIL Method.

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Step 2.1

Apply the distributive property.

$\frac{d}{d x} [{(8 - 5 x)}^{3} - 7 (8 (8 - 5 x) - 5 x (8 - 5 x)) + 3 (8 - 5 x) - 1]$

Step 2.2

Apply the distributive property.

$\frac{d}{d x} [{(8 - 5 x)}^{3} - 7 (8 \cdot 8 + 8 (- 5 x) - 5 x (8 - 5 x)) + 3 (8 - 5 x) - 1]$

Step 2.3

Apply the distributive property.

$\frac{d}{d x} [{(8 - 5 x)}^{3} - 7 (8 \cdot 8 + 8 (- 5 x) - 5 x \cdot 8 - 5 x (- 5 x)) + 3 (8 - 5 x) - 1]$

Step 3

Simplify and combine like terms.

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Step 3.1

Simplify each term.

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Step 3.1.1

Multiply $8$ by $8$ .

$\frac{d}{d x} [{(8 - 5 x)}^{3} - 7 (64 + 8 (- 5 x) - 5 x \cdot 8 - 5 x (- 5 x)) + 3 (8 - 5 x) - 1]$

Step 3.1.2

Multiply $- 5$ by $8$ .

$\frac{d}{d x} [{(8 - 5 x)}^{3} - 7 (64 - 40 x - 5 x \cdot 8 - 5 x (- 5 x)) + 3 (8 - 5 x) - 1]$

Step 3.1.3

Multiply $8$ by $- 5$ .

$\frac{d}{d x} [{(8 - 5 x)}^{3} - 7 (64 - 40 x - 40 x - 5 x (- 5 x)) + 3 (8 - 5 x) - 1]$

Step 3.1.4

Rewrite using the commutative property of multiplication.

$\frac{d}{d x} [{(8 - 5 x)}^{3} - 7 (64 - 40 x - 40 x - 5 \cdot - 5 x \cdot x) + 3 (8 - 5 x) - 1]$

Step 3.1.5

Multiply $x$ by $x$ by adding the exponents.

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Step 3.1.5.1

Move $x$ .

$\frac{d}{d x} [{(8 - 5 x)}^{3} - 7 (64 - 40 x - 40 x - 5 \cdot - 5 (x \cdot x)) + 3 (8 - 5 x) - 1]$

Step 3.1.5.2

Multiply $x$ by $x$ .

$\frac{d}{d x} [{(8 - 5 x)}^{3} - 7 (64 - 40 x - 40 x - 5 \cdot - 5 x^{2}) + 3 (8 - 5 x) - 1]$

Step 3.1.6

Multiply $- 5$ by $- 5$ .

$\frac{d}{d x} [{(8 - 5 x)}^{3} - 7 (64 - 40 x - 40 x + 25 x^{2}) + 3 (8 - 5 x) - 1]$

Step 3.2

Subtract $40 x$ from $- 40 x$ .

$\frac{d}{d x} [{(8 - 5 x)}^{3} - 7 (64 - 80 x + 25 x^{2}) + 3 (8 - 5 x) - 1]$

Step 4

By the Sum Rule, the derivative of ${(8 - 5 x)}^{3} - 7 (64 - 80 x + 25 x^{2}) + 3 (8 - 5 x) - 1$ with respect to $x$ is $\frac{d}{d x} [{(8 - 5 x)}^{3}] + \frac{d}{d x} [- 7 (64 - 80 x + 25 x^{2})] + \frac{d}{d x} [3 (8 - 5 x)] + \frac{d}{d x} [- 1]$ .

$\frac{d}{d x} [{(8 - 5 x)}^{3}] + \frac{d}{d x} [- 7 (64 - 80 x + 25 x^{2})] + \frac{d}{d x} [3 (8 - 5 x)] + \frac{d}{d x} [- 1]$

Step 5

Evaluate $\frac{d}{d x} [{(8 - 5 x)}^{3}]$ .

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Step 5.1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{3}$ and $g (x) = 8 - 5 x$ .

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Step 5.1.1

To apply the Chain Rule, set $u$ as $8 - 5 x$ .

$\frac{d}{d u} [u^{3}] \frac{d}{d x} [8 - 5 x] + \frac{d}{d x} [- 7 (64 - 80 x + 25 x^{2})] + \frac{d}{d x} [3 (8 - 5 x)] + \frac{d}{d x} [- 1]$

Step 5.1.2

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

$3 u^{2} \frac{d}{d x} [8 - 5 x] + \frac{d}{d x} [- 7 (64 - 80 x + 25 x^{2})] + \frac{d}{d x} [3 (8 - 5 x)] + \frac{d}{d x} [- 1]$

Step 5.1.3

Replace all occurrences of $u$ with $8 - 5 x$ .

$3 {(8 - 5 x)}^{2} \frac{d}{d x} [8 - 5 x] + \frac{d}{d x} [- 7 (64 - 80 x + 25 x^{2})] + \frac{d}{d x} [3 (8 - 5 x)] + \frac{d}{d x} [- 1]$

Step 5.2

By the Sum Rule, the derivative of $8 - 5 x$ with respect to $x$ is $\frac{d}{d x} [8] + \frac{d}{d x} [- 5 x]$ .

$3 {(8 - 5 x)}^{2} (\frac{d}{d x} [8] + \frac{d}{d x} [- 5 x]) + \frac{d}{d x} [- 7 (64 - 80 x + 25 x^{2})] + \frac{d}{d x} [3 (8 - 5 x)] + \frac{d}{d x} [- 1]$

Step 5.3

Since $8$ is constant with respect to $x$ , the derivative of $8$ with respect to $x$ is $0$ .

$3 {(8 - 5 x)}^{2} (0 + \frac{d}{d x} [- 5 x]) + \frac{d}{d x} [- 7 (64 - 80 x + 25 x^{2})] + \frac{d}{d x} [3 (8 - 5 x)] + \frac{d}{d x} [- 1]$

Step 5.4

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

$3 {(8 - 5 x)}^{2} (0 - 5 \frac{d}{d x} [x]) + \frac{d}{d x} [- 7 (64 - 80 x + 25 x^{2})] + \frac{d}{d x} [3 (8 - 5 x)] + \frac{d}{d x} [- 1]$

Step 5.5

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

$3 {(8 - 5 x)}^{2} (0 - 5 \cdot 1) + \frac{d}{d x} [- 7 (64 - 80 x + 25 x^{2})] + \frac{d}{d x} [3 (8 - 5 x)] + \frac{d}{d x} [- 1]$

Step 5.6

Multiply $- 5$ by $1$ .

$3 {(8 - 5 x)}^{2} (0 - 5) + \frac{d}{d x} [- 7 (64 - 80 x + 25 x^{2})] + \frac{d}{d x} [3 (8 - 5 x)] + \frac{d}{d x} [- 1]$

Step 5.7

Subtract $5$ from $0$ .

$3 {(8 - 5 x)}^{2} \cdot - 5 + \frac{d}{d x} [- 7 (64 - 80 x + 25 x^{2})] + \frac{d}{d x} [3 (8 - 5 x)] + \frac{d}{d x} [- 1]$

Step 5.8

Multiply $- 5$ by $3$ .

$- 15 {(8 - 5 x)}^{2} + \frac{d}{d x} [- 7 (64 - 80 x + 25 x^{2})] + \frac{d}{d x} [3 (8 - 5 x)] + \frac{d}{d x} [- 1]$

Step 6

Evaluate $\frac{d}{d x} [- 7 (64 - 80 x + 25 x^{2})]$ .

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Step 6.1

Since $- 7$ is constant with respect to $x$ , the derivative of $- 7 (64 - 80 x + 25 x^{2})$ with respect to $x$ is $- 7 \frac{d}{d x} [64 - 80 x + 25 x^{2}]$ .

$- 15 {(8 - 5 x)}^{2} - 7 \frac{d}{d x} [64 - 80 x + 25 x^{2}] + \frac{d}{d x} [3 (8 - 5 x)] + \frac{d}{d x} [- 1]$

Step 6.2

By the Sum Rule, the derivative of $64 - 80 x + 25 x^{2}$ with respect to $x$ is $\frac{d}{d x} [64] + \frac{d}{d x} [- 80 x] + \frac{d}{d x} [25 x^{2}]$ .

$- 15 {(8 - 5 x)}^{2} - 7 (\frac{d}{d x} [64] + \frac{d}{d x} [- 80 x] + \frac{d}{d x} [25 x^{2}]) + \frac{d}{d x} [3 (8 - 5 x)] + \frac{d}{d x} [- 1]$

Step 6.3

Since $64$ is constant with respect to $x$ , the derivative of $64$ with respect to $x$ is $0$ .

$- 15 {(8 - 5 x)}^{2} - 7 (0 + \frac{d}{d x} [- 80 x] + \frac{d}{d x} [25 x^{2}]) + \frac{d}{d x} [3 (8 - 5 x)] + \frac{d}{d x} [- 1]$

Step 6.4

Since $- 80$ is constant with respect to $x$ , the derivative of $- 80 x$ with respect to $x$ is $- 80 \frac{d}{d x} [x]$ .

$- 15 {(8 - 5 x)}^{2} - 7 (0 - 80 \frac{d}{d x} [x] + \frac{d}{d x} [25 x^{2}]) + \frac{d}{d x} [3 (8 - 5 x)] + \frac{d}{d x} [- 1]$

Step 6.5

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

$- 15 {(8 - 5 x)}^{2} - 7 (0 - 80 \cdot 1 + \frac{d}{d x} [25 x^{2}]) + \frac{d}{d x} [3 (8 - 5 x)] + \frac{d}{d x} [- 1]$

Step 6.6

Since $25$ is constant with respect to $x$ , the derivative of $25 x^{2}$ with respect to $x$ is $25 \frac{d}{d x} [x^{2}]$ .

$- 15 {(8 - 5 x)}^{2} - 7 (0 - 80 \cdot 1 + 25 \frac{d}{d x} [x^{2}]) + \frac{d}{d x} [3 (8 - 5 x)] + \frac{d}{d x} [- 1]$

Step 6.7

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

$- 15 {(8 - 5 x)}^{2} - 7 (0 - 80 \cdot 1 + 25 (2 x)) + \frac{d}{d x} [3 (8 - 5 x)] + \frac{d}{d x} [- 1]$

Step 6.8

Multiply $- 80$ by $1$ .

$- 15 {(8 - 5 x)}^{2} - 7 (0 - 80 + 25 (2 x)) + \frac{d}{d x} [3 (8 - 5 x)] + \frac{d}{d x} [- 1]$

Step 6.9

Subtract $80$ from $0$ .

$- 15 {(8 - 5 x)}^{2} - 7 (- 80 + 25 (2 x)) + \frac{d}{d x} [3 (8 - 5 x)] + \frac{d}{d x} [- 1]$

Step 6.10

Multiply $2$ by $25$ .

$- 15 {(8 - 5 x)}^{2} - 7 (- 80 + 50 x) + \frac{d}{d x} [3 (8 - 5 x)] + \frac{d}{d x} [- 1]$

Step 7

Evaluate $\frac{d}{d x} [3 (8 - 5 x)]$ .

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Step 7.1

Since $3$ is constant with respect to $x$ , the derivative of $3 (8 - 5 x)$ with respect to $x$ is $3 \frac{d}{d x} [8 - 5 x]$ .

$- 15 {(8 - 5 x)}^{2} - 7 (- 80 + 50 x) + 3 \frac{d}{d x} [8 - 5 x] + \frac{d}{d x} [- 1]$

Step 7.2

By the Sum Rule, the derivative of $8 - 5 x$ with respect to $x$ is $\frac{d}{d x} [8] + \frac{d}{d x} [- 5 x]$ .

$- 15 {(8 - 5 x)}^{2} - 7 (- 80 + 50 x) + 3 (\frac{d}{d x} [8] + \frac{d}{d x} [- 5 x]) + \frac{d}{d x} [- 1]$

Step 7.3

Since $8$ is constant with respect to $x$ , the derivative of $8$ with respect to $x$ is $0$ .

$- 15 {(8 - 5 x)}^{2} - 7 (- 80 + 50 x) + 3 (0 + \frac{d}{d x} [- 5 x]) + \frac{d}{d x} [- 1]$

Step 7.4

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

$- 15 {(8 - 5 x)}^{2} - 7 (- 80 + 50 x) + 3 (0 - 5 \frac{d}{d x} [x]) + \frac{d}{d x} [- 1]$

Step 7.5

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

$- 15 {(8 - 5 x)}^{2} - 7 (- 80 + 50 x) + 3 (0 - 5 \cdot 1) + \frac{d}{d x} [- 1]$

Step 7.6

Multiply $- 5$ by $1$ .

$- 15 {(8 - 5 x)}^{2} - 7 (- 80 + 50 x) + 3 (0 - 5) + \frac{d}{d x} [- 1]$

Step 7.7

Subtract $5$ from $0$ .

$- 15 {(8 - 5 x)}^{2} - 7 (- 80 + 50 x) + 3 \cdot - 5 + \frac{d}{d x} [- 1]$

Step 7.8

Multiply $3$ by $- 5$ .

$- 15 {(8 - 5 x)}^{2} - 7 (- 80 + 50 x) - 15 + \frac{d}{d x} [- 1]$

Step 8

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

$- 15 {(8 - 5 x)}^{2} - 7 (- 80 + 50 x) - 15 + 0$

Step 9

Simplify.

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Step 9.1

Apply the distributive property.

$- 15 {(8 - 5 x)}^{2} - 7 \cdot - 80 - 7 (50 x) - 15 + 0$

Step 9.2

Combine terms.

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Step 9.2.1

Multiply $- 7$ by $- 80$ .

$- 15 {(8 - 5 x)}^{2} + 560 - 7 (50 x) - 15 + 0$

Step 9.2.2

Multiply $50$ by $- 7$ .

$- 15 {(8 - 5 x)}^{2} + 560 - 350 x - 15 + 0$

Step 9.2.3

Subtract $15$ from $560$ .

$- 15 {(8 - 5 x)}^{2} - 350 x + 545 + 0$

Step 9.2.4

Add $- 15 {(8 - 5 x)}^{2} - 350 x + 545$ and $0$ .

$- 15 {(8 - 5 x)}^{2} - 350 x + 545$

Step 9.3

Simplify each term.

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Step 9.3.1

Rewrite ${(8 - 5 x)}^{2}$ as $(8 - 5 x) (8 - 5 x)$ .

$- 15 ((8 - 5 x) (8 - 5 x)) - 350 x + 545$

Step 9.3.2

Expand $(8 - 5 x) (8 - 5 x)$ using the FOIL Method.

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Step 9.3.2.1

Apply the distributive property.

$- 15 (8 (8 - 5 x) - 5 x (8 - 5 x)) - 350 x + 545$

Step 9.3.2.2

Apply the distributive property.

$- 15 (8 \cdot 8 + 8 (- 5 x) - 5 x (8 - 5 x)) - 350 x + 545$

Step 9.3.2.3

Apply the distributive property.

$- 15 (8 \cdot 8 + 8 (- 5 x) - 5 x \cdot 8 - 5 x (- 5 x)) - 350 x + 545$

Step 9.3.3

Simplify and combine like terms.

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Step 9.3.3.1

Simplify each term.

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Step 9.3.3.1.1

Multiply $8$ by $8$ .

$- 15 (64 + 8 (- 5 x) - 5 x \cdot 8 - 5 x (- 5 x)) - 350 x + 545$

Step 9.3.3.1.2

Multiply $- 5$ by $8$ .

$- 15 (64 - 40 x - 5 x \cdot 8 - 5 x (- 5 x)) - 350 x + 545$

Step 9.3.3.1.3

Multiply $8$ by $- 5$ .

$- 15 (64 - 40 x - 40 x - 5 x (- 5 x)) - 350 x + 545$

Step 9.3.3.1.4

Rewrite using the commutative property of multiplication.

$- 15 (64 - 40 x - 40 x - 5 \cdot - 5 x \cdot x) - 350 x + 545$

Step 9.3.3.1.5

Multiply $x$ by $x$ by adding the exponents.

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Step 9.3.3.1.5.1

Move $x$ .

$- 15 (64 - 40 x - 40 x - 5 \cdot - 5 (x \cdot x)) - 350 x + 545$

Step 9.3.3.1.5.2

Multiply $x$ by $x$ .

$- 15 (64 - 40 x - 40 x - 5 \cdot - 5 x^{2}) - 350 x + 545$

Step 9.3.3.1.6

Multiply $- 5$ by $- 5$ .

$- 15 (64 - 40 x - 40 x + 25 x^{2}) - 350 x + 545$

Step 9.3.3.2

Subtract $40 x$ from $- 40 x$ .

$- 15 (64 - 80 x + 25 x^{2}) - 350 x + 545$

Step 9.3.4

Apply the distributive property.

$- 15 \cdot 64 - 15 (- 80 x) - 15 (25 x^{2}) - 350 x + 545$

Step 9.3.5

Simplify.

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Step 9.3.5.1

Multiply $- 15$ by $64$ .

$- 960 - 15 (- 80 x) - 15 (25 x^{2}) - 350 x + 545$

Step 9.3.5.2

Multiply $- 80$ by $- 15$ .

$- 960 + 1200 x - 15 (25 x^{2}) - 350 x + 545$

Step 9.3.5.3

Multiply $25$ by $- 15$ .

$- 960 + 1200 x - 375 x^{2} - 350 x + 545$

Step 9.4

Add $- 960$ and $545$ .

$1200 x - 375 x^{2} - 350 x - 415$

Step 9.5

Subtract $350 x$ from $1200 x$ .

$- 375 x^{2} + 850 x - 415$