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(c) y = e-a sin(ax + 3). (e) y = tan(5x) 52+1 (d) y = a √a² + x²

(c) y = e-a sin(ax + 3). (e) y = tan(5x) 52+1 (d) y = a √a² + x²

Calculus: Early Transcendentals
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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Please answer these three parts and solve the derivative while showing the steps.

### Mathematical Expressions: **(c)** \( y = e^{-ax} \sin(ax + 3) \) This is a function involving an exponential decay factor \( e^{-ax} \) and a sinusoidal component \(\sin(ax + 3)\). The parameter \(a\) affects both the rate of decay and the frequency of oscillation. **(d)** \( y = \frac{a}{\sqrt{a^2 + x^2}} \) This expression describes a relationship where \(y\) is inversely related to the square root of the sum of squares of \(a\) and \(x\). This function can often be involved in situations modeling potential fields or distances in physics. **(e)** \( y = \frac{\tan(5x)}{5x + 1} \) This expression is a rational function where the numerator is the tangent of \(5x\) and the denominator is a linear function \(5x + 1\). The behavior of this function will be determined by the interplay of the periodic tangent function and the linear term.
Transcribed Image Text:### Mathematical Expressions: **(c)** \( y = e^{-ax} \sin(ax + 3) \) This is a function involving an exponential decay factor \( e^{-ax} \) and a sinusoidal component \(\sin(ax + 3)\). The parameter \(a\) affects both the rate of decay and the frequency of oscillation. **(d)** \( y = \frac{a}{\sqrt{a^2 + x^2}} \) This expression describes a relationship where \(y\) is inversely related to the square root of the sum of squares of \(a\) and \(x\). This function can often be involved in situations modeling potential fields or distances in physics. **(e)** \( y = \frac{\tan(5x)}{5x + 1} \) This expression is a rational function where the numerator is the tangent of \(5x\) and the denominator is a linear function \(5x + 1\). The behavior of this function will be determined by the interplay of the periodic tangent function and the linear term.
Expert Solution
Step 1: Formula used

(d/dx)(af(x)) = a(d/dx)(f(x)) , where a is constant. 

(u.v)' = u.v'+u'.v 

(u/v)' = (v.u' - u.v')/v2

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