Textbook Question
Find the derivatives of the functions in Exercises 19–40.
y = (4x + 3)⁴(x + 1)⁻³
3. Techniques of Differentiation
The Chain Rule
3:46 minutesProblem 25a
Textbook Question
[Technology Exercise]
Draining a tank It takes 12 hours to drain a storage tank by opening the valve at the bottom. The depth y of fluid in the tank t hours after the valve is opened is given by the formula
y = 6(1 - t/12)² m.
a. Find the rate dy/dt (m/h) at which the tank is draining at time t.
Verified step by step guidance1
Identify the given function for the depth of fluid in the tank: \( y = 6(1 - \frac{t}{12})^2 \). This function represents the depth of the fluid in meters as a function of time \( t \) in hours.
To find the rate at which the tank is draining, we need to compute the derivative of \( y \) with respect to \( t \), which is \( \frac{dy}{dt} \). This derivative will give us the rate of change of the fluid depth over time.
Apply the chain rule to differentiate \( y = 6(1 - \frac{t}{12})^2 \). The chain rule states that if you have a composite function \( f(g(t)) \), then \( \frac{d}{dt}[f(g(t))] = f'(g(t)) \cdot g'(t) \).
First, differentiate the outer function \( f(u) = 6u^2 \) with respect to \( u \), where \( u = 1 - \frac{t}{12} \). The derivative is \( f'(u) = 12u \).
Next, differentiate the inner function \( g(t) = 1 - \frac{t}{12} \) with respect to \( t \). The derivative is \( g'(t) = -\frac{1}{12} \). Combine these using the chain rule: \( \frac{dy}{dt} = 12(1 - \frac{t}{12}) \cdot (-\frac{1}{12}) \).
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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 represents the rate at which the function's value changes with respect to a change in its input. In this context, dy/dt is the derivative of the depth y with respect to time t, indicating how quickly the fluid level in the tank is decreasing over time.
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Derivatives
Chain Rule
The chain rule is a fundamental technique in calculus used to differentiate composite functions. It states that the derivative of a composite function is the derivative of the outer function evaluated at the inner function times the derivative of the inner function. This rule is essential for finding dy/dt when y is expressed as a function of t.
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Quadratic Function
A quadratic function is a polynomial function of degree two, typically in the form ax² + bx + c. In this problem, y = 6(1 - t/12)² is a quadratic function of t. Understanding its structure helps in applying the chain rule and finding the derivative, as it involves recognizing the inner function (1 - t/12) and its square.
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