Calculus Examples

$f (x) = \sin (3 x^{5})$

Step 1

Find the first derivative.

Tap for more steps...

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 $g (x) = 3 x^{5}$ .

Tap for more steps...

To apply the Chain Rule, set $u$ as $3 x^{5}$ .

$f' (x) = \frac{d}{d u} (\sin (u)) \frac{d}{d x} (3 x^{5})$

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

$f' (x) = \cos (u) \frac{d}{d x} (3 x^{5})$

Replace all occurrences of $u$ with $3 x^{5}$ .

$f' (x) = \cos (3 x^{5}) \frac{d}{d x} (3 x^{5})$

Differentiate.

Tap for more steps...

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

$f' (x) = \cos (3 x^{5}) (3 \frac{d}{d x} (x^{5}))$

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

$f' (x) = \cos (3 x^{5}) (3 (5 x^{4}))$

Simplify the expression.

Tap for more steps...

Multiply $5$ by $3$ .

$f' (x) = \cos (3 x^{5}) (15 x^{4})$

Reorder the factors of $\cos (3 x^{5}) (15 x^{4})$ .

$f' (x) = 15 x^{4} \cos (3 x^{5})$

Step 2

Find the second derivative.

Tap for more steps...

Since $15$ is constant with respect to $x$ , the derivative of $15 x^{4} \cos (3 x^{5})$ with respect to $x$ is $15 \frac{d}{d x} [x^{4} \cos (3 x^{5})]$ .

$f'' (x) = 15 \frac{d}{d x} (x^{4} \cos (3 x^{5}))$

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

$f'' (x) = 15 (x^{4} \frac{d}{d x} (\cos (3 x^{5})) + \cos (3 x^{5}) \frac{d}{d x} (x^{4}))$

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \cos (x)$ and $g (x) = 3 x^{5}$ .

Tap for more steps...

To apply the Chain Rule, set $u$ as $3 x^{5}$ .

$f'' (x) = 15 (x^{4} (\frac{d}{d u} (\cos (u)) \frac{d}{d x} (3 x^{5})) + \cos (3 x^{5}) \frac{d}{d x} (x^{4}))$

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

$f'' (x) = 15 (x^{4} (- \sin (u) \frac{d}{d x} (3 x^{5})) + \cos (3 x^{5}) \frac{d}{d x} (x^{4}))$

Replace all occurrences of $u$ with $3 x^{5}$ .

$f'' (x) = 15 (x^{4} (- \sin (3 x^{5}) \frac{d}{d x} (3 x^{5})) + \cos (3 x^{5}) \frac{d}{d x} (x^{4}))$

Differentiate.

Tap for more steps...

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

$f'' (x) = 15 (x^{4} (- \sin (3 x^{5}) (3 \frac{d}{d x} (x^{5}))) + \cos (3 x^{5}) \frac{d}{d x} (x^{4}))$

Multiply $3$ by $- 1$ .

$f'' (x) = 15 (x^{4} (- 3 \sin (3 x^{5}) \frac{d}{d x} x^{5}) + \cos (3 x^{5}) \frac{d}{d x} (x^{4}))$

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

$f'' (x) = 15 (x^{4} (- 3 \sin (3 x^{5}) (5 x^{4})) + \cos (3 x^{5}) \frac{d}{d x} (x^{4}))$

Multiply $5$ by $- 3$ .

$f'' (x) = 15 (x^{4} (- 15 \sin (3 x^{5}) x^{4}) + \cos (3 x^{5}) \frac{d}{d x} (x^{4}))$

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

Tap for more steps...

Move $x^{4}$ .

$f'' (x) = 15 (x^{4} x^{4} (- 15 \sin (3 x^{5})) + \cos (3 x^{5}) \frac{d}{d x} (x^{4}))$

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

$f'' (x) = 15 (x^{4 + 4} (- 15 \sin (3 x^{5})) + \cos (3 x^{5}) \frac{d}{d x} (x^{4}))$

Add $4$ and $4$ .

$f'' (x) = 15 (x^{8} (- 15 \sin (3 x^{5})) + \cos (3 x^{5}) \frac{d}{d x} (x^{4}))$

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

$f'' (x) = 15 (x^{8} (- 15 \sin (3 x^{5})) + \cos (3 x^{5}) (4 x^{3}))$

Simplify.

Tap for more steps...

Apply the distributive property.

$f'' (x) = 15 (x^{8} (- 15 \sin (3 x^{5}))) + 15 (\cos (3 x^{5}) (4 x^{3}))$

Combine terms.

Tap for more steps...

Multiply $- 15$ by $15$ .

$f'' (x) = - 225 (x^{8} (\sin (3 x^{5}))) + 15 (\cos (3 x^{5}) (4 x^{3}))$

Multiply $4$ by $15$ .

$f'' (x) = - 225 x^{8} \sin (3 x^{5}) + 60 \cos (3 x^{5}) x^{3}$

Reorder terms.

$f'' (x) = - 225 x^{8} \sin (3 x^{5}) + 60 x^{3} \cos (3 x^{5})$

Step 3

The second derivative of $f (x)$ with respect to $x$ is $- 225 x^{8} \sin (3 x^{5}) + 60 x^{3} \cos (3 x^{5})$ .

$- 225 x^{8} \sin (3 x^{5}) + 60 x^{3} \cos (3 x^{5})$