Calculus Examples

$f (x) = {(2 x - 5)}^{4} {(x^{2} + x + 1)}^{5}$

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 $f (x) = {(2 x - 5)}^{4}$ and $g (x) = {(x^{2} + x + 1)}^{5}$ .

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

Step 2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{5}$ and $g (x) = x^{2} + x + 1$ .

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Step 2.1

To apply the Chain Rule, set $u_{1}$ as $x^{2} + x + 1$ .

${(2 x - 5)}^{4} (\frac{d}{d u_{1}} [{u_{1}}^{5}] \frac{d}{d x} [x^{2} + x + 1]) + {(x^{2} + x + 1)}^{5} \frac{d}{d x} [{(2 x - 5)}^{4}]$

Step 2.2

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

${(2 x - 5)}^{4} (5 {u_{1}}^{4} \frac{d}{d x} [x^{2} + x + 1]) + {(x^{2} + x + 1)}^{5} \frac{d}{d x} [{(2 x - 5)}^{4}]$

Step 2.3

Replace all occurrences of $u_{1}$ with $x^{2} + x + 1$ .

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

Step 3

Differentiate.

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Step 3.1

Move $5$ to the left of ${(2 x - 5)}^{4}$ .

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

Step 3.2

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

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

Step 3.3

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

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

Step 3.4

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

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

Step 3.5

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

$5 {(2 x - 5)}^{4} {(x^{2} + x + 1)}^{4} (2 x + 1 + 0) + {(x^{2} + x + 1)}^{5} \frac{d}{d x} [{(2 x - 5)}^{4}]$

Step 3.6

Add $2 x + 1$ and $0$ .

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

Step 4

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{4}$ and $g (x) = 2 x - 5$ .

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Step 4.1

To apply the Chain Rule, set $u_{2}$ as $2 x - 5$ .

$5 {(2 x - 5)}^{4} {(x^{2} + x + 1)}^{4} (2 x + 1) + {(x^{2} + x + 1)}^{5} (\frac{d}{d u_{2}} [{u_{2}}^{4}] \frac{d}{d x} [2 x - 5])$

Step 4.2

Differentiate using the Power Rule which states that $\frac{d}{d u_{2}} [{u_{2}}^{n}]$ is $n {u_{2}}^{n - 1}$ where $n = 4$ .

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

Step 4.3

Replace all occurrences of $u_{2}$ with $2 x - 5$ .

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

Step 5

Differentiate.

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Step 5.1

Move $4$ to the left of ${(x^{2} + x + 1)}^{5}$ .

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

Step 5.2

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

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

Step 5.3

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

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

Step 5.4

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

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

Step 5.5

Multiply $2$ by $1$ .

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

Step 5.6

Since $- 5$ is constant with respect to $x$ , the derivative of $- 5$ with respect to $x$ is $0$ .

$5 {(2 x - 5)}^{4} {(x^{2} + x + 1)}^{4} (2 x + 1) + 4 {(x^{2} + x + 1)}^{5} {(2 x - 5)}^{3} (2 + 0)$

Step 5.7

Simplify the expression.

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Step 5.7.1

Add $2$ and $0$ .

$5 {(2 x - 5)}^{4} {(x^{2} + x + 1)}^{4} (2 x + 1) + 4 {(x^{2} + x + 1)}^{5} {(2 x - 5)}^{3} \cdot 2$

Step 5.7.2

Multiply $2$ by $4$ .

$5 {(2 x - 5)}^{4} {(x^{2} + x + 1)}^{4} (2 x + 1) + 8 {(x^{2} + x + 1)}^{5} {(2 x - 5)}^{3}$

Step 6

Simplify.

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Step 6.1

Factor ${(2 x - 5)}^{3} {(x^{2} + x + 1)}^{4}$ out of $5 {(2 x - 5)}^{4} {(x^{2} + x + 1)}^{4} (2 x + 1) + 8 {(x^{2} + x + 1)}^{5} {(2 x - 5)}^{3}$ .

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Step 6.1.1

Factor ${(2 x - 5)}^{3} {(x^{2} + x + 1)}^{4}$ out of $5 {(2 x - 5)}^{4} {(x^{2} + x + 1)}^{4} (2 x + 1)$ .

${(2 x - 5)}^{3} {(x^{2} + x + 1)}^{4} (5 (2 x - 5) (2 x + 1)) + 8 {(x^{2} + x + 1)}^{5} {(2 x - 5)}^{3}$

Step 6.1.2

Factor ${(2 x - 5)}^{3} {(x^{2} + x + 1)}^{4}$ out of $8 {(x^{2} + x + 1)}^{5} {(2 x - 5)}^{3}$ .

${(2 x - 5)}^{3} {(x^{2} + x + 1)}^{4} (5 (2 x - 5) (2 x + 1)) + {(2 x - 5)}^{3} {(x^{2} + x + 1)}^{4} (8 (x^{2} + x + 1))$

Step 6.1.3

Factor ${(2 x - 5)}^{3} {(x^{2} + x + 1)}^{4}$ out of ${(2 x - 5)}^{3} {(x^{2} + x + 1)}^{4} (5 (2 x - 5) (2 x + 1)) + {(2 x - 5)}^{3} {(x^{2} + x + 1)}^{4} (8 (x^{2} + x + 1))$ .

${(2 x - 5)}^{3} {(x^{2} + x + 1)}^{4} (5 (2 x - 5) (2 x + 1) + 8 (x^{2} + x + 1))$

Step 6.2

Reorder the factors of ${(2 x - 5)}^{3} {(x^{2} + x + 1)}^{4} (5 (2 x - 5) (2 x + 1) + 8 (x^{2} + x + 1))$ .

${(x^{2} + x + 1)}^{4} {(2 x - 5)}^{3} (5 (2 x - 5) (2 x + 1) + 8 (x^{2} + x + 1))$