### Calculus Problem: Derivatives of Composite Functions**Problem Statement:**For the pair of functions, find the derivative of the composite function \( f(t(x)) \) in terms of one input variable.Given:- \( f(t) = \frac{5}{\ln(t)} \)- \( t(x) = 5x^3 - 4x^2 + 9x \)\[\frac{df}{dx} = \, ?\]**Additional Materials:**- eBook### Mathematical Modeling: Average Length of the Outer EarThe average length of the outer ear for men can be modeled as:\[ f(x) = 0.89 + 0.495 \ln x \, \text{inches} \]Where \( x \) is aligned input so that \( x(a) = a + 10 \) and \( a \) is age, with data ranging from \( 0 < a \leq 70 \).**Tasks:**- Calculate the average ear length and the change of ear length for 25-year-old men.- Round your answers to three decimal places. Sure, here's a transcription suitable for an educational website:---**Topic: Calculating Ear Length and Growth Rate****Problem Statement:**The average length of the outer ear for men can be modeled by the equation:\[ f(x) = 0.89 + 0.495 \ln x \]where \( x \) represents age in years, and data is collected from \( 0 < x \leq 70 \).**Objective:**Calculate the average ear length and the rate of change of ear length for a 25-year-old man. Round your answers to three decimal places.1. **Calculate the Average Ear Length**: \[ f(x) \text{ at } x = 25 \text{:} \] \(\text{Average ear length} = \_\_\_\_\_ \) inches2. **Determine the Rate of Change**: \[ \left(\frac{df}{da}\right)_{a = 25} \] \(\text{Rate of change} = \_\_\_\_\_ \) inches per year**Interpretation:**Write a sentence of interpretation for these results. (Round your answers to three decimal places.)"The average outer ear length for 25-year-old males is approximately ______ inches and ______ by approximately ______ inches per year."**Additional Materials:**- eBook--- This transcription outlines the mathematical problem, objectives, and required interpretation for a clear educational understanding.

Name: ### Calculus Problem: Derivatives of Composite Functions**Problem Sta
Uploaded: 2024-03-20T06:47:22.000Z
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Description: ### Calculus Problem: Derivatives of Composite Functions**Problem Statement:**For the pair of functions, find the derivative of the composite function \( f(t(x)) \) in terms of one input variable.Given:- \( f(t) = \frac{5}{\ln(t)} \)- \( t(x) = 5x^3