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Compute the derivative

Compute the derivative

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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Question

Compute the derivative 

### Transcription for Educational Website The image contains the following mathematical functions: (a) \( g(t) = \ln(15t) \) This function represents a natural logarithm of the product of 15 and variable \( t \). (b) \( h(x) = e^{5x^2 + 3x} \) This function is an exponential function where the exponent is a quadratic expression, \( 5x^2 + 3x \). ### Explanation - **Natural Logarithm Function \( g(t) \):** - The function \( \ln(15t) \) is the logarithm of \( 15t \) with base \( e \) (Euler's number, approximately 2.718). It is defined for \( t > 0 \). The value of \( g(t) \) increases as \( t \) increases. - **Exponential Function \( h(x) \):** - This function \( e^{5x^2 + 3x} \) involves the exponentiation of the expression \( 5x^2 + 3x \). Exponential functions grow rapidly and are crucial in various applications such as population growth and compound interest calculations. Both functions are foundational in calculus and have various applications in scientific and engineering disciplines. Understanding their behavior helps in modeling complex real-world phenomena.
Transcribed Image Text:### Transcription for Educational Website The image contains the following mathematical functions: (a) \( g(t) = \ln(15t) \) This function represents a natural logarithm of the product of 15 and variable \( t \). (b) \( h(x) = e^{5x^2 + 3x} \) This function is an exponential function where the exponent is a quadratic expression, \( 5x^2 + 3x \). ### Explanation - **Natural Logarithm Function \( g(t) \):** - The function \( \ln(15t) \) is the logarithm of \( 15t \) with base \( e \) (Euler's number, approximately 2.718). It is defined for \( t > 0 \). The value of \( g(t) \) increases as \( t \) increases. - **Exponential Function \( h(x) \):** - This function \( e^{5x^2 + 3x} \) involves the exponentiation of the expression \( 5x^2 + 3x \). Exponential functions grow rapidly and are crucial in various applications such as population growth and compound interest calculations. Both functions are foundational in calculus and have various applications in scientific and engineering disciplines. Understanding their behavior helps in modeling complex real-world phenomena.
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