Calculus Examples

$y = \cos ({(π x)}^{4})$

Step 1

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

Tap for more steps...

Step 1.1

To apply the Chain Rule, set $u_{1}$ as ${(π x)}^{4}$ .

$\frac{d}{d u_{1}} [\cos (u_{1})] \frac{d}{d x} [{(π x)}^{4}]$

Step 1.2

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

$- \sin (u_{1}) \frac{d}{d x} [{(π x)}^{4}]$

Step 1.3

Replace all occurrences of $u_{1}$ with ${(π x)}^{4}$ .

$- \sin ({(π x)}^{4}) \frac{d}{d x} [{(π x)}^{4}]$

Step 2

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 $g (x) = π x$ .

Tap for more steps...

Step 2.1

To apply the Chain Rule, set $u_{2}$ as $π x$ .

$- \sin ({(π x)}^{4}) (\frac{d}{d u_{2}} [{u_{2}}^{4}] \frac{d}{d x} [π x])$

Step 2.2

Differentiate using the Power Rule which states that $\frac{d}{d u_{2}} [{u_{2}}^{n}]$ is $n {u_{2}}^{n - 1}$ where $n = 4$ .

$- \sin ({(π x)}^{4}) (4 {u_{2}}^{3} \frac{d}{d x} [π x])$

Step 2.3

Replace all occurrences of $u_{2}$ with $π x$ .

$- \sin ({(π x)}^{4}) (4 {(π x)}^{3} \frac{d}{d x} [π x])$

Step 3

Differentiate.

Tap for more steps...

Step 3.1

Multiply $4$ by $- 1$ .

$- 4 \sin ({(π x)}^{4}) ({(π x)}^{3} \frac{d}{d x} [π x])$

Step 3.2

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

$- 4 \sin ({(π x)}^{4}) ({(π x)}^{3} (π \frac{d}{d x} [x]))$

Step 3.3

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

$- 4 \sin ({(π x)}^{4}) ({(π x)}^{3} π \cdot 1)$

Step 3.4

Multiply ${(π x)}^{3}$ by $1$ .

$- 4 \sin ({(π x)}^{4}) ({(π x)}^{3} π)$

Step 4

Simplify.

Tap for more steps...

Step 4.1

Apply the product rule to $π x$ .

$- 4 \sin (π^{4} x^{4}) ({(π x)}^{3} π)$

Step 4.2

Apply the product rule to $π x$ .

$- 4 \sin (π^{4} x^{4}) (π^{3} x^{3} π)$

Step 4.3

Combine terms.

Tap for more steps...

Step 4.3.1

Raise $π$ to the power of $1$ .

$- 4 \sin (π^{4} x^{4}) (x^{3} (π^{3} π^{1}))$

Step 4.3.2

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

$- 4 \sin (π^{4} x^{4}) (x^{3} π^{3 + 1})$

Step 4.3.3

Add $3$ and $1$ .

$- 4 \sin (π^{4} x^{4}) (x^{3} π^{4})$

Step 4.4

Reorder the factors of $- 4 \sin (π^{4} x^{4}) (x^{3} π^{4})$ .

$- 4 π^{4} x^{3} \sin (π^{4} x^{4})$