Calculus Examples

$h (x) = 5 x^{2} {(9 x + 7)}^{4}$ , $x = 4$

Step 1

Find the derivative.

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Step 1.1

Since $5$ is constant with respect to $x$ , the derivative of $5 x^{2} {(9 x + 7)}^{4}$ with respect to $x$ is $5 \frac{d}{d x} [x^{2} {(9 x + 7)}^{4}]$ .

$5 \frac{d}{d x} [x^{2} {(9 x + 7)}^{4}]$

Step 1.2

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 $f (x) = x^{2}$ and $g (x) = {(9 x + 7)}^{4}$ .

$5 (x^{2} \frac{d}{d x} [{(9 x + 7)}^{4}] + {(9 x + 7)}^{4} \frac{d}{d x} [x^{2}])$

Step 1.3

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 $g (x) = 9 x + 7$ .

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Step 1.3.1

To apply the Chain Rule, set $u$ as $9 x + 7$ .

$5 (x^{2} (\frac{d}{d u} [u^{4}] \frac{d}{d x} [9 x + 7]) + {(9 x + 7)}^{4} \frac{d}{d x} [x^{2}])$

Step 1.3.2

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

$5 (x^{2} (4 u^{3} \frac{d}{d x} [9 x + 7]) + {(9 x + 7)}^{4} \frac{d}{d x} [x^{2}])$

Step 1.3.3

Replace all occurrences of $u$ with $9 x + 7$ .

$5 (x^{2} (4 {(9 x + 7)}^{3} \frac{d}{d x} [9 x + 7]) + {(9 x + 7)}^{4} \frac{d}{d x} [x^{2}])$

Step 1.4

Differentiate.

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Step 1.4.1

By the Sum Rule, the derivative of $9 x + 7$ with respect to $x$ is $\frac{d}{d x} [9 x] + \frac{d}{d x} [7]$ .

$5 (x^{2} (4 {(9 x + 7)}^{3} (\frac{d}{d x} [9 x] + \frac{d}{d x} [7])) + {(9 x + 7)}^{4} \frac{d}{d x} [x^{2}])$

Step 1.4.2

Since $9$ is constant with respect to $x$ , the derivative of $9 x$ with respect to $x$ is $9 \frac{d}{d x} [x]$ .

$5 (x^{2} (4 {(9 x + 7)}^{3} (9 \frac{d}{d x} [x] + \frac{d}{d x} [7])) + {(9 x + 7)}^{4} \frac{d}{d x} [x^{2}])$

Step 1.4.3

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

$5 (x^{2} (4 {(9 x + 7)}^{3} (9 \cdot 1 + \frac{d}{d x} [7])) + {(9 x + 7)}^{4} \frac{d}{d x} [x^{2}])$

Step 1.4.4

Multiply $9$ by $1$ .

$5 (x^{2} (4 {(9 x + 7)}^{3} (9 + \frac{d}{d x} [7])) + {(9 x + 7)}^{4} \frac{d}{d x} [x^{2}])$

Step 1.4.5

Since $7$ is constant with respect to $x$ , the derivative of $7$ with respect to $x$ is $0$ .

$5 (x^{2} (4 {(9 x + 7)}^{3} (9 + 0)) + {(9 x + 7)}^{4} \frac{d}{d x} [x^{2}])$

Step 1.4.6

Simplify the expression.

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Step 1.4.6.1

Add $9$ and $0$ .

$5 (x^{2} (4 {(9 x + 7)}^{3} \cdot 9) + {(9 x + 7)}^{4} \frac{d}{d x} [x^{2}])$

Step 1.4.6.2

Multiply $9$ by $4$ .

$5 (x^{2} (36 {(9 x + 7)}^{3}) + {(9 x + 7)}^{4} \frac{d}{d x} [x^{2}])$

Step 1.4.7

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

$5 (x^{2} (36 {(9 x + 7)}^{3}) + {(9 x + 7)}^{4} (2 x))$

Step 1.4.8

Move $2$ to the left of ${(9 x + 7)}^{4}$ .

$5 (x^{2} (36 {(9 x + 7)}^{3}) + 2 \cdot {(9 x + 7)}^{4} x)$

Step 1.5

Simplify.

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Step 1.5.1

Apply the distributive property.

$5 (x^{2} (36 {(9 x + 7)}^{3})) + 5 (2 {(9 x + 7)}^{4} x)$

Step 1.5.2

Multiply $36$ by $5$ .

$180 (x^{2} ({(9 x + 7)}^{3})) + 5 (2 {(9 x + 7)}^{4} x)$

Step 1.5.3

Multiply $2$ by $5$ .

$180 x^{2} {(9 x + 7)}^{3} + 10 ({(9 x + 7)}^{4} x)$

Step 1.5.4

Factor $10 x {(9 x + 7)}^{3}$ out of $180 x^{2} {(9 x + 7)}^{3} + 10 {(9 x + 7)}^{4} x$ .

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Step 1.5.4.1

Factor $10 x {(9 x + 7)}^{3}$ out of $180 x^{2} {(9 x + 7)}^{3}$ .

$10 x {(9 x + 7)}^{3} (18 x) + 10 {(9 x + 7)}^{4} x$

Step 1.5.4.2

Factor $10 x {(9 x + 7)}^{3}$ out of $10 {(9 x + 7)}^{4} x$ .

$10 x {(9 x + 7)}^{3} (18 x) + 10 x {(9 x + 7)}^{3} (9 x + 7)$

Step 1.5.4.3

Factor $10 x {(9 x + 7)}^{3}$ out of $10 x {(9 x + 7)}^{3} (18 x) + 10 x {(9 x + 7)}^{3} (9 x + 7)$ .

$10 x {(9 x + 7)}^{3} (18 x + 9 x + 7)$

Step 1.5.5

Add $18 x$ and $9 x$ .

$10 x {(9 x + 7)}^{3} (27 x + 7)$

Step 2

Replace the variable $x$ with $4$ in the expression.

$h' (4) = 10 (4) {(9 (4) + 7)}^{3} (27 (4) + 7)$

Step 3

Multiply $10$ by $4$ .

$h' (4) = 40 {(9 (4) + 7)}^{3} (27 (4) + 7)$

Step 4

Multiply $9$ by $4$ .

$h' (4) = 40 {(36 + 7)}^{3} (27 (4) + 7)$

Step 5

Add $36$ and $7$ .

$h' (4) = 40 \cdot (43^{3} (27 (4) + 7))$

Step 6

Raise $43$ to the power of $3$ .

$h' (4) = 40 \cdot (79507 (27 (4) + 7))$

Step 7

Multiply $40$ by $79507$ .

$h' (4) = 3180280 (27 (4) + 7)$

Step 8

Multiply $27$ by $4$ .

$h' (4) = 3180280 (108 + 7)$

Step 9

Add $108$ and $7$ .

$h' (4) = 3180280 \cdot 115$

Step 10

Multiply $3180280$ by $115$ .

$h' (4) = 365732200$