Explained: "Use Version 2 of the Chain Rule to..."

College
Calculus and Analysis
Differential Calculus
Derivatives

Jazmyn Holden

Answered question

2022-03-23

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

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

Answer & Explanation

Esteban Sloan

Beginner2022-03-24Added 21 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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