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

Kathleen Rausch

Answered question

2021-12-08

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

y = {\csc e}^{x}

Barbara Meeker

Beginner2021-12-09Added 38 answers

Step 1
The chain rule of two functions f(x) and g(x) is given by the following formula.

\frac{d}{d x} (f (g (x))) = f^{'} (g (x)) \cdot g^{'} (x)

Where g(x) is an inner function and f(x) is an outer function.
Step 2

I n y = {\csc e}^{x}

, the inner function

u = e^{x}

and the outer function

f (u) = \csc u

. First, differentiate

u = e^{x}

with respect to x.

\frac{d u}{d x} = \frac{d e^{x}}{d x}

= e^{x}

Now, differentiate

f (u) = \csc u

with respect to u.

f^{'} (u) = \frac{d \csc u}{d u} = - \csc u \cot u

Step 3
Use the chain rule to differentiate

y = {\csc e}^{x}

\frac{d y}{d x} = \frac{d}{d x} {\csc e}^{x}

= \frac{d}{d x} f (u)

= f^{'} (u) \cdot \frac{d u}{d x}

= - \csc u \cot u \cdot e^{x}

= - e^{x} {\csc e}^{x} {\cot e}^{x}

Hence,

\frac{d y}{d x} = - e^{x} {\csc e}^{x} {\cot e}^{x}

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