Calculus Examples

$y = \sin (9 x^{5})$

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 $f (x) = \sin (x)$ and $g (x) = 9 x^{5}$ .

Tap for more steps...

Step 1.1.1

To apply the Chain Rule, set $u$ as $9 x^{5}$ .

$\frac{d}{d u} [\sin (u)] \frac{d}{d x} [9 x^{5}]$

Step 1.1.2

The derivative of $\sin (u)$ with respect to $u$ is $\cos (u)$ .

$\cos (u) \frac{d}{d x} [9 x^{5}]$

Step 1.1.3

Replace all occurrences of $u$ with $9 x^{5}$ .

$\cos (9 x^{5}) \frac{d}{d x} [9 x^{5}]$

Step 1.2

Differentiate.

Tap for more steps...

Step 1.2.1

Since $9$ is constant with respect to $x$ , the derivative of $9 x^{5}$ with respect to $x$ is $9 \frac{d}{d x} [x^{5}]$ .

$\cos (9 x^{5}) (9 \frac{d}{d x} [x^{5}])$

Step 1.2.2

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

$\cos (9 x^{5}) (9 (5 x^{4}))$

Step 1.2.3

Simplify the expression.

Tap for more steps...

Step 1.2.3.1

Multiply $5$ by $9$ .

$\cos (9 x^{5}) (45 x^{4})$

Step 1.2.3.2

Reorder the factors of $\cos (9 x^{5}) (45 x^{4})$ .

$f' (x) = 45 x^{4} \cos (9 x^{5})$

Step 2

Find the second derivative.

Tap for more steps...

Step 2.1

Since $45$ is constant with respect to $x$ , the derivative of $45 x^{4} \cos (9 x^{5})$ with respect to $x$ is $45 \frac{d}{d x} [x^{4} \cos (9 x^{5})]$ .

$45 \frac{d}{d x} [x^{4} \cos (9 x^{5})]$

Step 2.2

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 $f (x) = x^{4}$ and $g (x) = \cos (9 x^{5})$ .

$45 (x^{4} \frac{d}{d x} [\cos (9 x^{5})] + \cos (9 x^{5}) \frac{d}{d x} [x^{4}])$

Step 2.3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \cos (x)$ and $g (x) = 9 x^{5}$ .

Tap for more steps...

Step 2.3.1

To apply the Chain Rule, set $u$ as $9 x^{5}$ .

$45 (x^{4} (\frac{d}{d u} [\cos (u)] \frac{d}{d x} [9 x^{5}]) + \cos (9 x^{5}) \frac{d}{d x} [x^{4}])$

Step 2.3.2

The derivative of $\cos (u)$ with respect to $u$ is $- \sin (u)$ .

$45 (x^{4} (- \sin (u) \frac{d}{d x} [9 x^{5}]) + \cos (9 x^{5}) \frac{d}{d x} [x^{4}])$

Step 2.3.3

Replace all occurrences of $u$ with $9 x^{5}$ .

$45 (x^{4} (- \sin (9 x^{5}) \frac{d}{d x} [9 x^{5}]) + \cos (9 x^{5}) \frac{d}{d x} [x^{4}])$

Step 2.4

Differentiate.

Tap for more steps...

Step 2.4.1

Since $9$ is constant with respect to $x$ , the derivative of $9 x^{5}$ with respect to $x$ is $9 \frac{d}{d x} [x^{5}]$ .

$45 (x^{4} (- \sin (9 x^{5}) (9 \frac{d}{d x} [x^{5}])) + \cos (9 x^{5}) \frac{d}{d x} [x^{4}])$

Step 2.4.2

Multiply $9$ by $- 1$ .

$45 (x^{4} (- 9 \sin (9 x^{5}) \frac{d}{d x} [x^{5}]) + \cos (9 x^{5}) \frac{d}{d x} [x^{4}])$

Step 2.4.3

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

$45 (x^{4} (- 9 \sin (9 x^{5}) (5 x^{4})) + \cos (9 x^{5}) \frac{d}{d x} [x^{4}])$

Step 2.4.4

Multiply $5$ by $- 9$ .

$45 (x^{4} (- 45 \sin (9 x^{5}) x^{4}) + \cos (9 x^{5}) \frac{d}{d x} [x^{4}])$

Step 2.5

Multiply $x^{4}$ by $x^{4}$ by adding the exponents.

Tap for more steps...

Step 2.5.1

Move $x^{4}$ .

$45 (x^{4} x^{4} (- 45 \sin (9 x^{5})) + \cos (9 x^{5}) \frac{d}{d x} [x^{4}])$

Step 2.5.2

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

$45 (x^{4 + 4} (- 45 \sin (9 x^{5})) + \cos (9 x^{5}) \frac{d}{d x} [x^{4}])$

Step 2.5.3

Add $4$ and $4$ .

$45 (x^{8} (- 45 \sin (9 x^{5})) + \cos (9 x^{5}) \frac{d}{d x} [x^{4}])$

Step 2.6

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

$45 (x^{8} (- 45 \sin (9 x^{5})) + \cos (9 x^{5}) (4 x^{3}))$

Step 2.7

Simplify.

Tap for more steps...

Step 2.7.1

Apply the distributive property.

$45 (x^{8} (- 45 \sin (9 x^{5}))) + 45 (\cos (9 x^{5}) (4 x^{3}))$

Step 2.7.2

Combine terms.

Tap for more steps...

Step 2.7.2.1

Multiply $- 45$ by $45$ .

$- 2025 (x^{8} (\sin (9 x^{5}))) + 45 (\cos (9 x^{5}) (4 x^{3}))$

Step 2.7.2.2

Multiply $4$ by $45$ .

$- 2025 x^{8} \sin (9 x^{5}) + 180 \cos (9 x^{5}) x^{3}$

Step 2.7.3

Reorder terms.

$f'' (x) = - 2025 x^{8} \sin (9 x^{5}) + 180 x^{3} \cos (9 x^{5})$

Step 3

Find the third derivative.

Tap for more steps...

Step 3.1

By the Sum Rule, the derivative of $- 2025 x^{8} \sin (9 x^{5}) + 180 x^{3} \cos (9 x^{5})$ with respect to $x$ is $\frac{d}{d x} [- 2025 x^{8} \sin (9 x^{5})] + \frac{d}{d x} [180 x^{3} \cos (9 x^{5})]$ .

$\frac{d}{d x} [- 2025 x^{8} \sin (9 x^{5})] + \frac{d}{d x} [180 x^{3} \cos (9 x^{5})]$

Step 3.2

Evaluate $\frac{d}{d x} [- 2025 x^{8} \sin (9 x^{5})]$ .

Tap for more steps...

Step 3.2.1

Since $- 2025$ is constant with respect to $x$ , the derivative of $- 2025 x^{8} \sin (9 x^{5})$ with respect to $x$ is $- 2025 \frac{d}{d x} [x^{8} \sin (9 x^{5})]$ .

$- 2025 \frac{d}{d x} [x^{8} \sin (9 x^{5})] + \frac{d}{d x} [180 x^{3} \cos (9 x^{5})]$

Step 3.2.2

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 $f (x) = x^{8}$ and $g (x) = \sin (9 x^{5})$ .

$- 2025 (x^{8} \frac{d}{d x} [\sin (9 x^{5})] + \sin (9 x^{5}) \frac{d}{d x} [x^{8}]) + \frac{d}{d x} [180 x^{3} \cos (9 x^{5})]$

Step 3.2.3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \sin (x)$ and $g (x) = 9 x^{5}$ .

Tap for more steps...

Step 3.2.3.1

To apply the Chain Rule, set $u_{1}$ as $9 x^{5}$ .

$- 2025 (x^{8} (\frac{d}{d u_{1}} [\sin (u_{1})] \frac{d}{d x} [9 x^{5}]) + \sin (9 x^{5}) \frac{d}{d x} [x^{8}]) + \frac{d}{d x} [180 x^{3} \cos (9 x^{5})]$

Step 3.2.3.2

The derivative of $\sin (u_{1})$ with respect to $u_{1}$ is $\cos (u_{1})$ .

$- 2025 (x^{8} (\cos (u_{1}) \frac{d}{d x} [9 x^{5}]) + \sin (9 x^{5}) \frac{d}{d x} [x^{8}]) + \frac{d}{d x} [180 x^{3} \cos (9 x^{5})]$

Step 3.2.3.3

Replace all occurrences of $u_{1}$ with $9 x^{5}$ .

$- 2025 (x^{8} (\cos (9 x^{5}) \frac{d}{d x} [9 x^{5}]) + \sin (9 x^{5}) \frac{d}{d x} [x^{8}]) + \frac{d}{d x} [180 x^{3} \cos (9 x^{5})]$

Step 3.2.4

Since $9$ is constant with respect to $x$ , the derivative of $9 x^{5}$ with respect to $x$ is $9 \frac{d}{d x} [x^{5}]$ .

$- 2025 (x^{8} (\cos (9 x^{5}) (9 \frac{d}{d x} [x^{5}])) + \sin (9 x^{5}) \frac{d}{d x} [x^{8}]) + \frac{d}{d x} [180 x^{3} \cos (9 x^{5})]$

Step 3.2.5

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

$- 2025 (x^{8} (\cos (9 x^{5}) (9 (5 x^{4}))) + \sin (9 x^{5}) \frac{d}{d x} [x^{8}]) + \frac{d}{d x} [180 x^{3} \cos (9 x^{5})]$

Step 3.2.6

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 8$ .

$- 2025 (x^{8} (\cos (9 x^{5}) (9 (5 x^{4}))) + \sin (9 x^{5}) (8 x^{7})) + \frac{d}{d x} [180 x^{3} \cos (9 x^{5})]$

Step 3.2.7

Multiply $5$ by $9$ .

$- 2025 (x^{8} (\cos (9 x^{5}) (45 x^{4})) + \sin (9 x^{5}) (8 x^{7})) + \frac{d}{d x} [180 x^{3} \cos (9 x^{5})]$

Step 3.2.8

Multiply $x^{8}$ by $x^{4}$ by adding the exponents.

Tap for more steps...

Step 3.2.8.1

Move $x^{4}$ .

$- 2025 (x^{4} x^{8} (\cos (9 x^{5}) \cdot (45)) + \sin (9 x^{5}) (8 x^{7})) + \frac{d}{d x} [180 x^{3} \cos (9 x^{5})]$

Step 3.2.8.2

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

$- 2025 (x^{4 + 8} (\cos (9 x^{5}) \cdot (45)) + \sin (9 x^{5}) (8 x^{7})) + \frac{d}{d x} [180 x^{3} \cos (9 x^{5})]$

Step 3.2.8.3

Add $4$ and $8$ .

$- 2025 (x^{12} (\cos (9 x^{5}) \cdot (45)) + \sin (9 x^{5}) (8 x^{7})) + \frac{d}{d x} [180 x^{3} \cos (9 x^{5})]$

Step 3.2.9

Move $45$ to the left of $\cos (9 x^{5})$ .

$- 2025 (x^{12} (45 \cos (9 x^{5})) + \sin (9 x^{5}) (8 x^{7})) + \frac{d}{d x} [180 x^{3} \cos (9 x^{5})]$

Step 3.3

Evaluate $\frac{d}{d x} [180 x^{3} \cos (9 x^{5})]$ .

Tap for more steps...

Step 3.3.1

Since $180$ is constant with respect to $x$ , the derivative of $180 x^{3} \cos (9 x^{5})$ with respect to $x$ is $180 \frac{d}{d x} [x^{3} \cos (9 x^{5})]$ .

$- 2025 (x^{12} (45 \cos (9 x^{5})) + \sin (9 x^{5}) (8 x^{7})) + 180 \frac{d}{d x} [x^{3} \cos (9 x^{5})]$

Step 3.3.2

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 $f (x) = x^{3}$ and $g (x) = \cos (9 x^{5})$ .

$- 2025 (x^{12} (45 \cos (9 x^{5})) + \sin (9 x^{5}) (8 x^{7})) + 180 (x^{3} \frac{d}{d x} [\cos (9 x^{5})] + \cos (9 x^{5}) \frac{d}{d x} [x^{3}])$

Step 3.3.3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \cos (x)$ and $g (x) = 9 x^{5}$ .

Tap for more steps...

Step 3.3.3.1

To apply the Chain Rule, set $u_{2}$ as $9 x^{5}$ .

$- 2025 (x^{12} (45 \cos (9 x^{5})) + \sin (9 x^{5}) (8 x^{7})) + 180 (x^{3} (\frac{d}{d u_{2}} [\cos (u_{2})] \frac{d}{d x} [9 x^{5}]) + \cos (9 x^{5}) \frac{d}{d x} [x^{3}])$

Step 3.3.3.2

The derivative of $\cos (u_{2})$ with respect to $u_{2}$ is $- \sin (u_{2})$ .

$- 2025 (x^{12} (45 \cos (9 x^{5})) + \sin (9 x^{5}) (8 x^{7})) + 180 (x^{3} (- \sin (u_{2}) \frac{d}{d x} [9 x^{5}]) + \cos (9 x^{5}) \frac{d}{d x} [x^{3}])$

Step 3.3.3.3

Replace all occurrences of $u_{2}$ with $9 x^{5}$ .

$- 2025 (x^{12} (45 \cos (9 x^{5})) + \sin (9 x^{5}) (8 x^{7})) + 180 (x^{3} (- \sin (9 x^{5}) \frac{d}{d x} [9 x^{5}]) + \cos (9 x^{5}) \frac{d}{d x} [x^{3}])$

Step 3.3.4

Since $9$ is constant with respect to $x$ , the derivative of $9 x^{5}$ with respect to $x$ is $9 \frac{d}{d x} [x^{5}]$ .

$- 2025 (x^{12} (45 \cos (9 x^{5})) + \sin (9 x^{5}) (8 x^{7})) + 180 (x^{3} (- \sin (9 x^{5}) (9 \frac{d}{d x} [x^{5}])) + \cos (9 x^{5}) \frac{d}{d x} [x^{3}])$

Step 3.3.5

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

$- 2025 (x^{12} (45 \cos (9 x^{5})) + \sin (9 x^{5}) (8 x^{7})) + 180 (x^{3} (- \sin (9 x^{5}) (9 (5 x^{4}))) + \cos (9 x^{5}) \frac{d}{d x} [x^{3}])$

Step 3.3.6

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

$- 2025 (x^{12} (45 \cos (9 x^{5})) + \sin (9 x^{5}) (8 x^{7})) + 180 (x^{3} (- \sin (9 x^{5}) (9 (5 x^{4}))) + \cos (9 x^{5}) (3 x^{2}))$

Step 3.3.7

Multiply $5$ by $9$ .

$- 2025 (x^{12} (45 \cos (9 x^{5})) + \sin (9 x^{5}) (8 x^{7})) + 180 (x^{3} (- \sin (9 x^{5}) (45 x^{4})) + \cos (9 x^{5}) (3 x^{2}))$

Step 3.3.8

Multiply $45$ by $- 1$ .

$- 2025 (x^{12} (45 \cos (9 x^{5})) + \sin (9 x^{5}) (8 x^{7})) + 180 (x^{3} (- 45 \sin (9 x^{5}) x^{4}) + \cos (9 x^{5}) (3 x^{2}))$

Step 3.3.9

Multiply $x^{3}$ by $x^{4}$ by adding the exponents.

Tap for more steps...

Step 3.3.9.1

Move $x^{4}$ .

$- 2025 (x^{12} (45 \cos (9 x^{5})) + \sin (9 x^{5}) (8 x^{7})) + 180 (x^{4} x^{3} (- 45 \sin (9 x^{5})) + \cos (9 x^{5}) (3 x^{2}))$

Step 3.3.9.2

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

$- 2025 (x^{12} (45 \cos (9 x^{5})) + \sin (9 x^{5}) (8 x^{7})) + 180 (x^{4 + 3} (- 45 \sin (9 x^{5})) + \cos (9 x^{5}) (3 x^{2}))$

Step 3.3.9.3

Add $4$ and $3$ .

$- 2025 (x^{12} (45 \cos (9 x^{5})) + \sin (9 x^{5}) (8 x^{7})) + 180 (x^{7} (- 45 \sin (9 x^{5})) + \cos (9 x^{5}) (3 x^{2}))$

Step 3.4

Simplify.

Tap for more steps...

Step 3.4.1

Apply the distributive property.

$- 2025 (x^{12} (45 \cos (9 x^{5}))) - 2025 (\sin (9 x^{5}) (8 x^{7})) + 180 (x^{7} (- 45 \sin (9 x^{5})) + \cos (9 x^{5}) (3 x^{2}))$

Step 3.4.2

Apply the distributive property.

$- 2025 (x^{12} (45 \cos (9 x^{5}))) - 2025 (\sin (9 x^{5}) (8 x^{7})) + 180 (x^{7} (- 45 \sin (9 x^{5}))) + 180 (\cos (9 x^{5}) (3 x^{2}))$

Step 3.4.3

Combine terms.

Tap for more steps...

Step 3.4.3.1

Multiply $45$ by $- 2025$ .

$- 91125 (x^{12} (\cos (9 x^{5}))) - 2025 (\sin (9 x^{5}) (8 x^{7})) + 180 (x^{7} (- 45 \sin (9 x^{5}))) + 180 (\cos (9 x^{5}) (3 x^{2}))$

Step 3.4.3.2

Multiply $8$ by $- 2025$ .

$- 91125 x^{12} \cos (9 x^{5}) - 16200 (\sin (9 x^{5}) (x^{7})) + 180 (x^{7} (- 45 \sin (9 x^{5}))) + 180 (\cos (9 x^{5}) (3 x^{2}))$

Step 3.4.3.3

Multiply $- 45$ by $180$ .

$- 91125 x^{12} \cos (9 x^{5}) - 16200 \sin (9 x^{5}) x^{7} - 8100 (x^{7} (\sin (9 x^{5}))) + 180 (\cos (9 x^{5}) (3 x^{2}))$

Step 3.4.3.4

Multiply $3$ by $180$ .

$- 91125 x^{12} \cos (9 x^{5}) - 16200 \sin (9 x^{5}) x^{7} - 8100 x^{7} \sin (9 x^{5}) + 540 (\cos (9 x^{5}) (x^{2}))$

Step 3.4.3.5

Subtract $8100 x^{7} \sin (9 x^{5})$ from $- 16200 \sin (9 x^{5}) x^{7}$ .

Tap for more steps...

Step 3.4.3.5.1

Move $\sin (9 x^{5})$ .

$- 91125 x^{12} \cos (9 x^{5}) - 16200 x^{7} \sin (9 x^{5}) - 8100 x^{7} \sin (9 x^{5}) + 540 \cos (9 x^{5}) x^{2}$

Step 3.4.3.5.2

Subtract $8100 x^{7} \sin (9 x^{5})$ from $- 16200 x^{7} \sin (9 x^{5})$ .

$- 91125 x^{12} \cos (9 x^{5}) - 24300 x^{7} \sin (9 x^{5}) + 540 \cos (9 x^{5}) x^{2}$

Step 3.4.4

Reorder terms.

$f''' (x) = - 91125 x^{12} \cos (9 x^{5}) - 24300 x^{7} \sin (9 x^{5}) + 540 x^{2} \cos (9 x^{5})$