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

College
Calculus and Analysis
Differential Calculus
Derivatives

BolkowN

Answered question

2021-10-13

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

y = e^{\tan t}

Answer & Explanation

svartmaleJ

Skilled2021-10-14Added 92 answers

Step 1
Given function:
$y = e^{\tan t}$
Differentiate the above function w.r.t 't'
$\frac{d y}{d t} = \frac{d}{d t} e^{\tan t}$
The function $y = e^{\tan t}$ is the composition $f (g (t))$ of two functions $f (u) = e^{u} and g (t) = \tan t$ .
Step 2
Apply the chain rule
$\frac{d}{d t} [f (g (t))] = \frac{d}{d u} f (u) \frac{d}{d t} g (t)$
Here, $f (u) = e^{u} and g (t) = \tan t$
So, we get
$\frac{d y}{d t} = \frac{d}{d u} e^{u} \frac{d}{d t} \tan t$
$\frac{d y}{d t} = e^{u} \times \sec^{2} t$
Substitute the value of $u = \tan (t)$ , we get
$\frac{d y}{d t} = e^{\tan t} \times \sec^{2} t$

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