Textbook Question
Find the derivative of the following functions.
y = x sin x
3. Techniques of Differentiation
Product and Quotient Rules
3:59 minutesProblem 96
Textbook Question
{Use of Tech} Beak length The length of the culmen (the upper ridge of a bird’s bill) of a t-week-old Indian spotted owlet is modeled by the function L(t)=11.94 / 1 + 4e^−1.65t, where L is measured in millimeters.
a. Find L′(1) and interpret the meaning of this value.
Verified step by step guidance1
First, identify the function given: L(t) = \frac{11.94}{1 + 4e^{-1.65t}}. This is a rational function where the numerator is a constant and the denominator is a function of t.
To find L'(t), we need to differentiate L(t) with respect to t. This requires using the quotient rule for differentiation, which states that if you have a function \frac{u(t)}{v(t)}, its derivative is \frac{u'(t)v(t) - u(t)v'(t)}{(v(t))^2}.
In our case, u(t) = 11.94 and v(t) = 1 + 4e^{-1.65t}. The derivative of u(t), u'(t), is 0 because it is a constant. For v(t), apply the chain rule to find v'(t). The derivative of 4e^{-1.65t} is -6.6e^{-1.65t} (using the chain rule).
Substitute these derivatives into the quotient rule formula: L'(t) = \frac{0 \cdot (1 + 4e^{-1.65t}) - 11.94 \cdot (-6.6e^{-1.65t})}{(1 + 4e^{-1.65t})^2}. Simplify the expression to find L'(t).
Finally, evaluate L'(1) by substituting t = 1 into the derivative expression. This value represents the rate of change of the culmen length at t = 1 week, indicating how quickly the length is increasing at that specific time.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Derivative
The derivative of a function measures the rate at which the function's value changes as its input changes. In this context, L'(t) represents the instantaneous rate of change of the culmen length L with respect to time t. Calculating L'(1) will provide insight into how quickly the beak length is growing at t = 1 week.
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Exponential Function
The function L(t) includes an exponential term, e^(-1.65t), which influences the growth behavior of the beak length over time. Exponential functions are characterized by their rapid increase or decrease, depending on the sign of the exponent. Understanding this concept is crucial for interpreting how the beak length evolves as the owlet ages.
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Interpretation of Derivatives
Interpreting the value of a derivative involves understanding its practical significance in a real-world context. For L'(1), this means analyzing what the calculated rate of change indicates about the growth of the owlet's beak at one week old, such as whether it is growing rapidly, slowly, or not at all, which can have implications for the bird's development.
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