"Use Version 2 of the Chain Rule to..." was answered by our experts

Rui Baldwin

Answered question

2021-10-04

Use Version 2 of the Chain Rule to calculate the derivatives of the following functions.

y = \cos^{4} 0 + \sin^{4} 0

Answer & Explanation

Raheem Donnelly

Skilled2021-10-05Added 75 answers

Step 1
Given function

y = \cos^{4} 0 + \sin^{4} 0

The version 2 of the chain rule is

\frac{d y}{d x} = \frac{d}{d x} (f (g (x)))

=f'(g(x))*g'(x)
Here let

f (0) = 0^{4}

g (0) = \cos 0

h (0) = \sin 0

Step 2
The derivative will be

\frac{d y}{d 0} = \frac{d}{d 0} (f (g (0)) + f (h (0)))

= \frac{d}{d 0} (f (g (0))) + \frac{d}{d 0} (f (h (0)))

=f'(g(0))*g'(0)+f'(h(0))*h'(0)

= 4 \cos^{3} 0 - \sin 0 + 4 \sin^{3} 0 \cos 0

= 4 (\sin^{3} 0 \cos 0 - \cos^{3} 0 \sin 0)

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