### Calculus: Determining the First DerivativeTo enhance your understanding of calculus, practice using any derivative rule to determine the first derivative of the following functions. Simplification of the result is not necessary. #### Problem 1:\[ h(x) = e^{-x} + e^{\pi} - \log_3(x) \]#### Problem 2: \[ p(x) = x \sin(x) \]**Instructions:**1. Identify and apply the appropriate derivative rules to each function.2. Recall that the derivative of \(e^u\) where \(u\) is a function of \(x\) is \(e^u \cdot u'\).3. For logarithms, the derivative of \(\log_b(x)\) is \(\frac{1}{x \ln(b)}\).4. Utilize the product rule for derivatives when necessary, specifically for functions where products of two functions are involved. The product rule states: \((uv)' = u'v + uv'\). Explaining in detail:- For \( c) \), apply the chain rule and the knowledge of basic derivative formulas for exponential and logarithmic functions.- For \( d) \), recognize that this function requires the use of the product rule since it is the product of \( x \) and \(\sin(x)\). Apply the product rule accordingly.By working through these exercises, you will gain a deeper understanding and become more proficient at differentiating a variety of functions.

Answered: Image /qna-images/answer/ccb6fdec-1818-4e1c-baa7-d4fb2193196b.jpg

Answered: c)hcx)-c*+ e - log: ()

c)hcx)-c*+ e - log: ()

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...

Similar questions

a) log, (16)+ log, (4)
State the restriction(s) on the variable x in log(2x-5) – log (4 - x).
Exercise 4. Log(1 – iV3)² = a. 2Log(1 – iV3) b. Log(1+ iv3) c. 2Log(1 + iv3) d. None of these
q15b-
The domain of f(x) = log,(1 - x) is Select one: O a ( -1,1) O b[1,1] O c. (0, 00) O d. [1, 00)
1. (1 point) Use properties of logarithms to simplify the following expressions. a.) log.(43)= b.) eIn() = c.) In(e)= d.) log(1) = 2. (1 point) Use properties of logarithms to combine each of the expressions below into one logarithm. a.) log3(x-1)-log3(x+1) b.) log2(x) + log2(y) - log2(z) 3. (1 point) Use properties of logarithms to split each of the expressions below into two logarithms. Remember to simplify all exponents. a.) log(x³y2) b.) In ( 4. (1 point) Consider the logarithmic expression: log,(100) a.) Estimate the value of this expression. b.) Show how to use the change of base formula to rewrite this expression using the common log or the natural log. Then use a calculator to approximate the expression to 2 decimal places.
The solution(s) of the equation 22x - 2* = 20 is (are): A) x= 5. x = log2(5) and x =- log2(4). x = log2(5). B) C) D) = log2 (20). E) log2 (20) and x = log2(21). A) B) c) D) E)
Q5) Find (x) a) log,(27) + log,(x) = 3.5 b) 2log,(x) – log3(x+ 6) = 1 c) log.(x) + log.(x + 5) = 2 d) log,(2x – 3) – log,(x + 2) = 1
Pls help ASAP.
Pls helpp ASAP
Thanks