Calculus Examples

$y = {(7 + \frac{4}{x})}^{4}$

Step 1

Find the first derivative.

Tap for more steps...

Step 1.1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{4}$ and $g (x) = 7 + \frac{4}{x}$ .

Tap for more steps...

Step 1.1.1

To apply the Chain Rule, set $u$ as $7 + \frac{4}{x}$ .

$\frac{d}{d u} [u^{4}] \frac{d}{d x} [7 + \frac{4}{x}]$

Step 1.1.2

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

$4 u^{3} \frac{d}{d x} [7 + \frac{4}{x}]$

Step 1.1.3

Replace all occurrences of $u$ with $7 + \frac{4}{x}$ .

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

Step 1.2

Differentiate.

Tap for more steps...

Step 1.2.1

By the Sum Rule, the derivative of $7 + \frac{4}{x}$ with respect to $x$ is $\frac{d}{d x} [7] + \frac{d}{d x} [\frac{4}{x}]$ .

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

Step 1.2.2

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

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

Step 1.2.3

Add $0$ and $\frac{d}{d x} [\frac{4}{x}]$ .

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

Step 1.2.4

Since $4$ is constant with respect to $x$ , the derivative of $\frac{4}{x}$ with respect to $x$ is $4 \frac{d}{d x} [\frac{1}{x}]$ .

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

Step 1.2.5

Simplify the expression.

Tap for more steps...

Step 1.2.5.1

Multiply $4$ by $4$ .

$16 {(7 + \frac{4}{x})}^{3} \frac{d}{d x} [\frac{1}{x}]$

Step 1.2.5.2

Rewrite $\frac{1}{x}$ as $x^{- 1}$ .

$16 {(7 + \frac{4}{x})}^{3} \frac{d}{d x} [x^{- 1}]$

Step 1.2.6

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

$16 {(7 + \frac{4}{x})}^{3} (- x^{- 2})$

Step 1.2.7

Multiply $- 1$ by $16$ .

$- 16 {(7 + \frac{4}{x})}^{3} x^{- 2}$

Step 1.3

Simplify.

Tap for more steps...

Step 1.3.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$- 16 {(7 + \frac{4}{x})}^{3} \frac{1}{x^{2}}$

Step 1.3.2

Combine terms.

Tap for more steps...

Step 1.3.2.1

Combine $\frac{1}{x^{2}}$ and $- 16$ .

$\frac{- 16}{x^{2}} {(7 + \frac{4}{x})}^{3}$

Step 1.3.2.2

Combine $\frac{- 16}{x^{2}}$ and ${(7 + \frac{4}{x})}^{3}$ .

$\frac{- 16 {(7 + \frac{4}{x})}^{3}}{x^{2}}$

Step 1.3.2.3

Move the negative in front of the fraction.

$- \frac{16 {(7 + \frac{4}{x})}^{3}}{x^{2}}$

Step 1.3.3

Simplify the numerator.

Tap for more steps...

Step 1.3.3.1

To write $7$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

$- \frac{16 {(\frac{7 x}{x} + \frac{4}{x})}^{3}}{x^{2}}$

Step 1.3.3.2

Combine the numerators over the common denominator.

$- \frac{16 {(\frac{7 x + 4}{x})}^{3}}{x^{2}}$

Step 1.3.3.3

Apply the product rule to $\frac{7 x + 4}{x}$ .

$- \frac{16 \frac{{(7 x + 4)}^{3}}{x^{3}}}{x^{2}}$

Step 1.3.4

Combine $16$ and $\frac{{(7 x + 4)}^{3}}{x^{3}}$ .

$- \frac{\frac{16 {(7 x + 4)}^{3}}{x^{3}}}{x^{2}}$

Step 1.3.5

Multiply the numerator by the reciprocal of the denominator.

$- (\frac{16 {(7 x + 4)}^{3}}{x^{3}} \cdot \frac{1}{x^{2}})$

Step 1.3.6

Combine.

$- \frac{16 {(7 x + 4)}^{3} \cdot 1}{x^{3} x^{2}}$

Step 1.3.7

Multiply $x^{3}$ by $x^{2}$ by adding the exponents.

Tap for more steps...

Step 1.3.7.1

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$- \frac{16 {(7 x + 4)}^{3} \cdot 1}{x^{3 + 2}}$

Step 1.3.7.2

Add $3$ and $2$ .

$- \frac{16 {(7 x + 4)}^{3} \cdot 1}{x^{5}}$

Step 1.3.8

Multiply $16 {(7 x + 4)}^{3}$ by $1$ .

$f' (x) = - \frac{16 {(7 x + 4)}^{3}}{x^{5}}$

Step 2

Find the second derivative.

Tap for more steps...

Step 2.1

Since $- 16$ is constant with respect to $x$ , the derivative of $- \frac{16 {(7 x + 4)}^{3}}{x^{5}}$ with respect to $x$ is $- 16 \frac{d}{d x} [\frac{{(7 x + 4)}^{3}}{x^{5}}]$ .

$- 16 \frac{d}{d x} [\frac{{(7 x + 4)}^{3}}{x^{5}}]$

Step 2.2

Differentiate using the Quotient Rule which states that $\frac{d}{d x} [\frac{f (x)}{g (x)}]$ is $\frac{g (x) \frac{d}{d x} [f (x)] - f (x) \frac{d}{d x} [g (x)]}{{g (x)}^{2}}$ where $f (x) = {(7 x + 4)}^{3}$ and $g (x) = x^{5}$ .

$- 16 \frac{x^{5} \frac{d}{d x} [{(7 x + 4)}^{3}] - {(7 x + 4)}^{3} \frac{d}{d x} [x^{5}]}{{(x^{5})}^{2}}$

Step 2.3

Multiply the exponents in ${(x^{5})}^{2}$ .

Tap for more steps...

Step 2.3.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$- 16 \frac{x^{5} \frac{d}{d x} [{(7 x + 4)}^{3}] - {(7 x + 4)}^{3} \frac{d}{d x} [x^{5}]}{x^{5 \cdot 2}}$

Step 2.3.2

Multiply $5$ by $2$ .

$- 16 \frac{x^{5} \frac{d}{d x} [{(7 x + 4)}^{3}] - {(7 x + 4)}^{3} \frac{d}{d x} [x^{5}]}{x^{10}}$

Step 2.4

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{3}$ and $g (x) = 7 x + 4$ .

Tap for more steps...

Step 2.4.1

To apply the Chain Rule, set $u$ as $7 x + 4$ .

$- 16 \frac{x^{5} (\frac{d}{d u} [u^{3}] \frac{d}{d x} [7 x + 4]) - {(7 x + 4)}^{3} \frac{d}{d x} [x^{5}]}{x^{10}}$

Step 2.4.2

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

$- 16 \frac{x^{5} (3 u^{2} \frac{d}{d x} [7 x + 4]) - {(7 x + 4)}^{3} \frac{d}{d x} [x^{5}]}{x^{10}}$

Step 2.4.3

Replace all occurrences of $u$ with $7 x + 4$ .

$- 16 \frac{x^{5} (3 {(7 x + 4)}^{2} \frac{d}{d x} [7 x + 4]) - {(7 x + 4)}^{3} \frac{d}{d x} [x^{5}]}{x^{10}}$

Step 2.5

Differentiate.

Tap for more steps...

Step 2.5.1

By the Sum Rule, the derivative of $7 x + 4$ with respect to $x$ is $\frac{d}{d x} [7 x] + \frac{d}{d x} [4]$ .

$- 16 \frac{x^{5} (3 {(7 x + 4)}^{2} (\frac{d}{d x} [7 x] + \frac{d}{d x} [4])) - {(7 x + 4)}^{3} \frac{d}{d x} [x^{5}]}{x^{10}}$

Step 2.5.2

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

$- 16 \frac{x^{5} (3 {(7 x + 4)}^{2} (7 \frac{d}{d x} [x] + \frac{d}{d x} [4])) - {(7 x + 4)}^{3} \frac{d}{d x} [x^{5}]}{x^{10}}$

Step 2.5.3

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

$- 16 \frac{x^{5} (3 {(7 x + 4)}^{2} (7 \cdot 1 + \frac{d}{d x} [4])) - {(7 x + 4)}^{3} \frac{d}{d x} [x^{5}]}{x^{10}}$

Step 2.5.4

Multiply $7$ by $1$ .

$- 16 \frac{x^{5} (3 {(7 x + 4)}^{2} (7 + \frac{d}{d x} [4])) - {(7 x + 4)}^{3} \frac{d}{d x} [x^{5}]}{x^{10}}$

Step 2.5.5

Since $4$ is constant with respect to $x$ , the derivative of $4$ with respect to $x$ is $0$ .

$- 16 \frac{x^{5} (3 {(7 x + 4)}^{2} (7 + 0)) - {(7 x + 4)}^{3} \frac{d}{d x} [x^{5}]}{x^{10}}$

Step 2.5.6

Simplify the expression.

Tap for more steps...

Step 2.5.6.1

Add $7$ and $0$ .

$- 16 \frac{x^{5} (3 {(7 x + 4)}^{2} \cdot 7) - {(7 x + 4)}^{3} \frac{d}{d x} [x^{5}]}{x^{10}}$

Step 2.5.6.2

Multiply $7$ by $3$ .

$- 16 \frac{x^{5} (21 {(7 x + 4)}^{2}) - {(7 x + 4)}^{3} \frac{d}{d x} [x^{5}]}{x^{10}}$

Step 2.5.7

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

$- 16 \frac{x^{5} (21 {(7 x + 4)}^{2}) - {(7 x + 4)}^{3} (5 x^{4})}{x^{10}}$

Step 2.5.8

Combine fractions.

Tap for more steps...

Step 2.5.8.1

Multiply $5$ by $- 1$ .

$- 16 \frac{x^{5} (21 {(7 x + 4)}^{2}) - 5 {(7 x + 4)}^{3} x^{4}}{x^{10}}$

Step 2.5.8.2

Combine $- 16$ and $\frac{x^{5} (21 {(7 x + 4)}^{2}) - 5 {(7 x + 4)}^{3} x^{4}}{x^{10}}$ .

$\frac{- 16 (x^{5} (21 {(7 x + 4)}^{2}) - 5 {(7 x + 4)}^{3} x^{4})}{x^{10}}$

Step 2.5.8.3

Move the negative in front of the fraction.

$- \frac{16 (x^{5} (21 {(7 x + 4)}^{2}) - 5 {(7 x + 4)}^{3} x^{4})}{x^{10}}$

Step 2.6

Simplify.

Tap for more steps...

Step 2.6.1

Apply the distributive property.

$- \frac{16 (x^{5} (21 {(7 x + 4)}^{2})) + 16 (- 5 {(7 x + 4)}^{3} x^{4})}{x^{10}}$

Step 2.6.2

Simplify the numerator.

Tap for more steps...

Step 2.6.2.1

Factor $16 x^{4} {(7 x + 4)}^{2}$ out of $16 x^{5} (21 {(7 x + 4)}^{2}) + 16 (- 5 {(7 x + 4)}^{3} x^{4})$ .

Tap for more steps...

Step 2.6.2.1.1

Factor $16 x^{4} {(7 x + 4)}^{2}$ out of $16 x^{5} (21 {(7 x + 4)}^{2})$ .

$- \frac{16 x^{4} {(7 x + 4)}^{2} (x \cdot (21)) + 16 (- 5 {(7 x + 4)}^{3} x^{4})}{x^{10}}$

Step 2.6.2.1.2

Factor $16 x^{4} {(7 x + 4)}^{2}$ out of $16 (- 5 {(7 x + 4)}^{3} x^{4})$ .

$- \frac{16 x^{4} {(7 x + 4)}^{2} (x \cdot (21)) + 16 x^{4} {(7 x + 4)}^{2} (- 5 (7 x + 4))}{x^{10}}$

Step 2.6.2.1.3

Factor $16 x^{4} {(7 x + 4)}^{2}$ out of $16 x^{4} {(7 x + 4)}^{2} (x \cdot (21)) + 16 x^{4} {(7 x + 4)}^{2} (- 5 (7 x + 4))$ .

$- \frac{16 x^{4} {(7 x + 4)}^{2} (x \cdot (21) - 5 (7 x + 4))}{x^{10}}$

Step 2.6.2.2

Move $21$ to the left of $x$ .

$- \frac{16 x^{4} {(7 x + 4)}^{2} (21 x - 5 (7 x + 4))}{x^{10}}$

Step 2.6.2.3

Apply the distributive property.

$- \frac{16 x^{4} {(7 x + 4)}^{2} (21 x - 5 (7 x) - 5 \cdot 4)}{x^{10}}$

Step 2.6.2.4

Multiply $7$ by $- 5$ .

$- \frac{16 x^{4} {(7 x + 4)}^{2} (21 x - 35 x - 5 \cdot 4)}{x^{10}}$

Step 2.6.2.5

Multiply $- 5$ by $4$ .

$- \frac{16 x^{4} {(7 x + 4)}^{2} (21 x - 35 x - 20)}{x^{10}}$

Step 2.6.2.6

Subtract $35 x$ from $21 x$ .

$- \frac{16 x^{4} {(7 x + 4)}^{2} (- 14 x - 20)}{x^{10}}$

Step 2.6.2.7

Factor $2$ out of $- 14 x - 20$ .

Tap for more steps...

Step 2.6.2.7.1

Factor $2$ out of $- 14 x$ .

$- \frac{16 x^{4} {(7 x + 4)}^{2} (2 (- 7 x) - 20)}{x^{10}}$

Step 2.6.2.7.2

Factor $2$ out of $- 20$ .

$- \frac{16 x^{4} {(7 x + 4)}^{2} (2 (- 7 x) + 2 (- 10))}{x^{10}}$

Step 2.6.2.7.3

Factor $2$ out of $2 (- 7 x) + 2 (- 10)$ .

$- \frac{16 x^{4} {(7 x + 4)}^{2} (2 (- 7 x - 10))}{x^{10}}$

$- \frac{16 x^{4} {(7 x + 4)}^{2} \cdot 2 (- 7 x - 10)}{x^{10}}$

Step 2.6.2.8

Multiply $2$ by $16$ .

$- \frac{32 x^{4} {(7 x + 4)}^{2} (- 7 x - 10)}{x^{10}}$

Step 2.6.3

Cancel the common factor of $x^{4}$ and $x^{10}$ .

Tap for more steps...

Step 2.6.3.1

Factor $x^{4}$ out of $32 x^{4} {(7 x + 4)}^{2} (- 7 x - 10)$ .

$- \frac{x^{4} (32 {(7 x + 4)}^{2} (- 7 x - 10))}{x^{10}}$

Step 2.6.3.2

Cancel the common factors.

Tap for more steps...

Step 2.6.3.2.1

Factor $x^{4}$ out of $x^{10}$ .

$- \frac{x^{4} (32 {(7 x + 4)}^{2} (- 7 x - 10))}{x^{4} x^{6}}$

Step 2.6.3.2.2

Cancel the common factor.

$- \frac{x^{4} (32 {(7 x + 4)}^{2} (- 7 x - 10))}{x^{4} x^{6}}$

Step 2.6.3.2.3

Rewrite the expression.

$- \frac{32 {(7 x + 4)}^{2} (- 7 x - 10)}{x^{6}}$

Step 2.6.4

Factor $- 1$ out of $- 7 x$ .

$- \frac{32 {(7 x + 4)}^{2} (- (7 x) - 10)}{x^{6}}$

Step 2.6.5

Rewrite $- 10$ as $- 1 (10)$ .

$- \frac{32 {(7 x + 4)}^{2} (- (7 x) - 1 (10))}{x^{6}}$

Step 2.6.6

Factor $- 1$ out of $- (7 x) - 1 (10)$ .

$- \frac{32 {(7 x + 4)}^{2} (- (7 x + 10))}{x^{6}}$

Step 2.6.7

Rewrite $- (7 x + 10)$ as $- 1 (7 x + 10)$ .

$- \frac{32 {(7 x + 4)}^{2} (- 1 (7 x + 10))}{x^{6}}$

Step 2.6.8

Move the negative in front of the fraction.

$- - \frac{(32 {(7 x + 4)}^{2}) (7 x + 10)}{x^{6}}$

Step 2.6.9

Multiply $- 1$ by $- 1$ .

$1 \frac{(32 {(7 x + 4)}^{2}) (7 x + 10)}{x^{6}}$

Step 2.6.10

Multiply $\frac{(32 {(7 x + 4)}^{2}) (7 x + 10)}{x^{6}}$ by $1$ .

$\frac{(32 {(7 x + 4)}^{2}) (7 x + 10)}{x^{6}}$

Step 2.6.11

Reorder factors in $\frac{(32 {(7 x + 4)}^{2}) (7 x + 10)}{x^{6}}$ .

$f'' (x) = \frac{32 {(7 x + 4)}^{2} (7 x + 10)}{x^{6}}$