Find the derivative of a trigonometric function. **Problem 45:**Find the function $ g(t) = \sqrt[4]{t} + 6 \csc t $.This mathematical expression defines a function $ g(t) $ which is the sum of two terms:1. **$\sqrt[4]{t}$:** This represents the fourth root of $ t $, which is a type of radical expression.2. **$6 \csc t$:** This term is 6 times the cosecant of $ t $, where cosecant is the trigonometric function, the reciprocal of sine. This function may be studied in a context involving analysis or graphing of equations with both algebraic and trigonometric components.

Name: Find the derivative of a trigonometric function. **Problem 45:**Find t
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Description: Find the derivative of a trigonometric function. **Problem 45:**Find the function $ g(t) = \sqrt[4]{t} + 6 \csc t $.This mathematical expression defines a function $ g(t) $ which is the sum of two terms:1. **$\sqrt[4]{t}$:** This represents the

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Calculus: Early Transcendentals

8th Edition

ISBN:9781285741550

Author:James Stewart

Publisher:James Stewart

Chapter1: Functions And Models

Section: Chapter Questions

Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...

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Find the derivative of a trigonometric function.

**Problem 45:**

Find the function $ g(t) = \sqrt[4]{t} + 6 \csc t $.

This mathematical expression defines a function $ g(t) $ which is the sum of two terms:

1. **$\sqrt[4]{t}$:** This represents the fourth root of $ t $, which is a type of radical expression.

2. **$6 \csc t$:** This term is 6 times the cosecant of $ t $, where cosecant is the trigonometric function, the reciprocal of sine.

This function may be studied in a context involving analysis or graphing of equations with both algebraic and trigonometric components.

Expert Solution

Step 1

We write g(t) using exponent first, then differentiate both sides with respect to t.

$g (t) = \sqrt[4]{t} + 6 \csc (t) g (t) = t^{\frac{1}{4}} + 6 \csc (t) g' (t) = \frac{d}{dt} (t^{\frac{1}{4}}) + \frac{d}{dt} (6 \csc (t)) g' (t) = \frac{1}{4} t^{\frac{1}{4} - 1} + 6 (- \csc (t) \cot (t)) g' (t) = \frac{1}{4} t^{- \frac{3}{4}} - 6 \csc (t) \cot (t) g' (t) = \frac{1}{4 t^{\frac{3}{4}}} - 6 \csc (t) \cot (t)$

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