Calculus Examples

$y = \cos^{4} (x) (1 - 2 t)$

Step 1

Since $1 - 2 t$ is constant with respect to $x$ , the derivative of $\cos^{4} (x) (1 - 2 t)$ with respect to $x$ is $(1 - 2 t) \frac{d}{d x} [\cos^{4} (x)]$ .

$(1 - 2 t) \frac{d}{d x} [\cos^{4} (x)]$

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 $g (x) = \cos (x)$ .

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Step 2.1

To apply the Chain Rule, set $u$ as $\cos (x)$ .

$(1 - 2 t) (\frac{d}{d u} [u^{4}] \frac{d}{d x} [\cos (x)])$

Step 2.2

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

$(1 - 2 t) (4 u^{3} \frac{d}{d x} [\cos (x)])$

Step 2.3

Replace all occurrences of $u$ with $\cos (x)$ .

$(1 - 2 t) (4 \cos^{3} (x) \frac{d}{d x} [\cos (x)])$

Step 3

Move $4$ to the left of $1 - 2 t$ .

$4 \cdot (1 - 2 t) \cos^{3} (x) \frac{d}{d x} [\cos (x)]$

Step 4

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

$4 (1 - 2 t) \cos^{3} (x) (- \sin (x))$

Step 5

Multiply $- 1$ by $4$ .

$- 4 (1 - 2 t) \cos^{3} (x) \sin (x)$

Step 6

Simplify.

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Step 6.1

Apply the distributive property.

$(- 4 \cdot 1 - 4 (- 2 t)) \cos^{3} (x) \sin (x)$

Step 6.2

Apply the distributive property.

$(- 4 \cdot 1 \cos^{3} (x) - 4 (- 2 t) \cos^{3} (x)) \sin (x)$

Step 6.3

Apply the distributive property.

$- 4 \cdot 1 \cos^{3} (x) \sin (x) - 4 (- 2 t) \cos^{3} (x) \sin (x)$

Step 6.4

Combine terms.

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Step 6.4.1

Multiply $- 4$ by $1$ .

$- 4 \cos^{3} (x) \sin (x) - 4 (- 2 t) \cos^{3} (x) \sin (x)$

Step 6.4.2

Multiply $- 2$ by $- 4$ .

$- 4 \cos^{3} (x) \sin (x) + 8 t \cos^{3} (x) \sin (x)$