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

Tony Mccarthy

Answered question

2022-03-24

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

y = e^{\tan t}

Answer & Explanation

Avery Maxwell

Beginner2022-03-25Added 13 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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