Calculus Examples

$y = {(3 x - 5)}^{2} {(5 - x^{5})}^{3}$

Step 1

Rewrite ${(3 x - 5)}^{2}$ as $(3 x - 5) (3 x - 5)$ .

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

Step 2

Expand $(3 x - 5) (3 x - 5)$ using the FOIL Method.

Tap for more steps...

Step 2.1

Apply the distributive property.

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

Step 2.2

Apply the distributive property.

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

Step 2.3

Apply the distributive property.

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

Step 3

Simplify and combine like terms.

Tap for more steps...

Step 3.1

Simplify each term.

Tap for more steps...

Step 3.1.1

Rewrite using the commutative property of multiplication.

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

Step 3.1.2

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

Tap for more steps...

Step 3.1.2.1

Move $x$ .

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

Step 3.1.2.2

Multiply $x$ by $x$ .

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

Step 3.1.3

Multiply $3$ by $3$ .

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

Step 3.1.4

Multiply $- 5$ by $3$ .

$\frac{d}{d x} [(9 x^{2} - 15 x - 5 (3 x) - 5 \cdot - 5) {(5 - x^{5})}^{3}]$

Step 3.1.5

Multiply $3$ by $- 5$ .

$\frac{d}{d x} [(9 x^{2} - 15 x - 15 x - 5 \cdot - 5) {(5 - x^{5})}^{3}]$

Step 3.1.6

Multiply $- 5$ by $- 5$ .

$\frac{d}{d x} [(9 x^{2} - 15 x - 15 x + 25) {(5 - x^{5})}^{3}]$

Step 3.2

Subtract $15 x$ from $- 15 x$ .

$\frac{d}{d x} [(9 x^{2} - 30 x + 25) {(5 - x^{5})}^{3}]$

Step 4

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 $f (x) = 9 x^{2} - 30 x + 25$ and $g (x) = {(5 - x^{5})}^{3}$ .

$(9 x^{2} - 30 x + 25) \frac{d}{d x} [{(5 - x^{5})}^{3}] + {(5 - x^{5})}^{3} \frac{d}{d x} [9 x^{2} - 30 x + 25]$

Step 5

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 $g (x) = 5 - x^{5}$ .

Tap for more steps...

Step 5.1

To apply the Chain Rule, set $u$ as $5 - x^{5}$ .

$(9 x^{2} - 30 x + 25) (\frac{d}{d u} [u^{3}] \frac{d}{d x} [5 - x^{5}]) + {(5 - x^{5})}^{3} \frac{d}{d x} [9 x^{2} - 30 x + 25]$

Step 5.2

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

$(9 x^{2} - 30 x + 25) (3 u^{2} \frac{d}{d x} [5 - x^{5}]) + {(5 - x^{5})}^{3} \frac{d}{d x} [9 x^{2} - 30 x + 25]$

Step 5.3

Replace all occurrences of $u$ with $5 - x^{5}$ .

$(9 x^{2} - 30 x + 25) (3 {(5 - x^{5})}^{2} \frac{d}{d x} [5 - x^{5}]) + {(5 - x^{5})}^{3} \frac{d}{d x} [9 x^{2} - 30 x + 25]$

Step 6

Differentiate.

Tap for more steps...

Step 6.1

Move $3$ to the left of $9 x^{2} - 30 x + 25$ .

$3 \cdot (9 x^{2} - 30 x + 25) {(5 - x^{5})}^{2} \frac{d}{d x} [5 - x^{5}] + {(5 - x^{5})}^{3} \frac{d}{d x} [9 x^{2} - 30 x + 25]$

Step 6.2

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

$3 (9 x^{2} - 30 x + 25) {(5 - x^{5})}^{2} (\frac{d}{d x} [5] + \frac{d}{d x} [- x^{5}]) + {(5 - x^{5})}^{3} \frac{d}{d x} [9 x^{2} - 30 x + 25]$

Step 6.3

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

$3 (9 x^{2} - 30 x + 25) {(5 - x^{5})}^{2} (0 + \frac{d}{d x} [- x^{5}]) + {(5 - x^{5})}^{3} \frac{d}{d x} [9 x^{2} - 30 x + 25]$

Step 6.4

Add $0$ and $\frac{d}{d x} [- x^{5}]$ .

$3 (9 x^{2} - 30 x + 25) {(5 - x^{5})}^{2} \frac{d}{d x} [- x^{5}] + {(5 - x^{5})}^{3} \frac{d}{d x} [9 x^{2} - 30 x + 25]$

Step 6.5

Since $- 1$ is constant with respect to $x$ , the derivative of $- x^{5}$ with respect to $x$ is $- \frac{d}{d x} [x^{5}]$ .

$3 (9 x^{2} - 30 x + 25) {(5 - x^{5})}^{2} (- \frac{d}{d x} [x^{5}]) + {(5 - x^{5})}^{3} \frac{d}{d x} [9 x^{2} - 30 x + 25]$

Step 6.6

Multiply $- 1$ by $3$ .

$- 3 (9 x^{2} - 30 x + 25) {(5 - x^{5})}^{2} \frac{d}{d x} [x^{5}] + {(5 - x^{5})}^{3} \frac{d}{d x} [9 x^{2} - 30 x + 25]$

Step 6.7

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

$- 3 (9 x^{2} - 30 x + 25) {(5 - x^{5})}^{2} (5 x^{4}) + {(5 - x^{5})}^{3} \frac{d}{d x} [9 x^{2} - 30 x + 25]$

Step 6.8

Multiply $5$ by $- 3$ .

$- 15 (9 x^{2} - 30 x + 25) {(5 - x^{5})}^{2} x^{4} + {(5 - x^{5})}^{3} \frac{d}{d x} [9 x^{2} - 30 x + 25]$

Step 6.9

By the Sum Rule, the derivative of $9 x^{2} - 30 x + 25$ with respect to $x$ is $\frac{d}{d x} [9 x^{2}] + \frac{d}{d x} [- 30 x] + \frac{d}{d x} [25]$ .

$- 15 (9 x^{2} - 30 x + 25) {(5 - x^{5})}^{2} x^{4} + {(5 - x^{5})}^{3} (\frac{d}{d x} [9 x^{2}] + \frac{d}{d x} [- 30 x] + \frac{d}{d x} [25])$

Step 6.10

Since $9$ is constant with respect to $x$ , the derivative of $9 x^{2}$ with respect to $x$ is $9 \frac{d}{d x} [x^{2}]$ .

$- 15 (9 x^{2} - 30 x + 25) {(5 - x^{5})}^{2} x^{4} + {(5 - x^{5})}^{3} (9 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 30 x] + \frac{d}{d x} [25])$

Step 6.11

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

$- 15 (9 x^{2} - 30 x + 25) {(5 - x^{5})}^{2} x^{4} + {(5 - x^{5})}^{3} (9 (2 x) + \frac{d}{d x} [- 30 x] + \frac{d}{d x} [25])$

Step 6.12

Multiply $2$ by $9$ .

$- 15 (9 x^{2} - 30 x + 25) {(5 - x^{5})}^{2} x^{4} + {(5 - x^{5})}^{3} (18 x + \frac{d}{d x} [- 30 x] + \frac{d}{d x} [25])$

Step 6.13

Since $- 30$ is constant with respect to $x$ , the derivative of $- 30 x$ with respect to $x$ is $- 30 \frac{d}{d x} [x]$ .

$- 15 (9 x^{2} - 30 x + 25) {(5 - x^{5})}^{2} x^{4} + {(5 - x^{5})}^{3} (18 x - 30 \frac{d}{d x} [x] + \frac{d}{d x} [25])$

Step 6.14

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

$- 15 (9 x^{2} - 30 x + 25) {(5 - x^{5})}^{2} x^{4} + {(5 - x^{5})}^{3} (18 x - 30 \cdot 1 + \frac{d}{d x} [25])$

Step 6.15

Multiply $- 30$ by $1$ .

$- 15 (9 x^{2} - 30 x + 25) {(5 - x^{5})}^{2} x^{4} + {(5 - x^{5})}^{3} (18 x - 30 + \frac{d}{d x} [25])$

Step 6.16

Since $25$ is constant with respect to $x$ , the derivative of $25$ with respect to $x$ is $0$ .

$- 15 (9 x^{2} - 30 x + 25) {(5 - x^{5})}^{2} x^{4} + {(5 - x^{5})}^{3} (18 x - 30 + 0)$

Step 6.17

Add $18 x - 30$ and $0$ .

$- 15 (9 x^{2} - 30 x + 25) {(5 - x^{5})}^{2} x^{4} + {(5 - x^{5})}^{3} (18 x - 30)$

Step 7

Simplify.

Tap for more steps...

Step 7.1

Apply the distributive property.

$(- 15 (9 x^{2}) - 15 (- 30 x) - 15 \cdot 25) {(5 - x^{5})}^{2} x^{4} + {(5 - x^{5})}^{3} (18 x - 30)$

Step 7.2

Multiply $9$ by $- 15$ .

$(- 135 x^{2} - 15 (- 30 x) - 15 \cdot 25) {(5 - x^{5})}^{2} x^{4} + {(5 - x^{5})}^{3} (18 x - 30)$

Step 7.3

Multiply $- 30$ by $- 15$ .

$(- 135 x^{2} + 450 x - 15 \cdot 25) {(5 - x^{5})}^{2} x^{4} + {(5 - x^{5})}^{3} (18 x - 30)$

Step 7.4

Multiply $- 15$ by $25$ .

$(- 135 x^{2} + 450 x - 375) {(5 - x^{5})}^{2} x^{4} + {(5 - x^{5})}^{3} (18 x - 30)$

Step 7.5

Factor ${(5 - x^{5})}^{2}$ out of $(- 135 x^{2} + 450 x - 375) {(5 - x^{5})}^{2} x^{4} + {(5 - x^{5})}^{3} (18 x - 30)$ .

Tap for more steps...

Step 7.5.1

Factor ${(5 - x^{5})}^{2}$ out of $(- 135 x^{2} + 450 x - 375) {(5 - x^{5})}^{2} x^{4}$ .

${(5 - x^{5})}^{2} ((- 135 x^{2} + 450 x - 375) x^{4}) + {(5 - x^{5})}^{3} (18 x - 30)$

Step 7.5.2

Factor ${(5 - x^{5})}^{2}$ out of ${(5 - x^{5})}^{3} (18 x - 30)$ .

${(5 - x^{5})}^{2} ((- 135 x^{2} + 450 x - 375) x^{4}) + {(5 - x^{5})}^{2} ((5 - x^{5}) (18 x - 30))$

Step 7.5.3

Factor ${(5 - x^{5})}^{2}$ out of ${(5 - x^{5})}^{2} ((- 135 x^{2} + 450 x - 375) x^{4}) + {(5 - x^{5})}^{2} ((5 - x^{5}) (18 x - 30))$ .

${(5 - x^{5})}^{2} ((- 135 x^{2} + 450 x - 375) x^{4} + (5 - x^{5}) (18 x - 30))$