Multiple Choice
Identify the open intervals on which the function is increasing or decreasing.
6. Derivatives of Inverse, Exponential, & Logarithmic Functions
Derivatives of Exponential & Logarithmic Functions
3:51 minutesProblem 3.9.44
Textbook Question
15–48. Derivatives Find the derivative of the following functions.
P = 40/1+2^-t
Verified step by step guidance1
Step 1: Identify the function P(t) = \frac{40}{1 + 2^{-t}}. This is a rational function where the numerator is a constant and the denominator is a function of t.
Step 2: Apply the quotient rule for derivatives, 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}. Here, u(t) = 40 and v(t) = 1 + 2^{-t}.
Step 3: Calculate u'(t). Since u(t) = 40, which is a constant, its derivative u'(t) = 0.
Step 4: Calculate v'(t). The function v(t) = 1 + 2^{-t} involves an exponential term. The derivative of 2^{-t} with respect to t is -2^{-t} \ln(2), using the chain rule.
Step 5: Substitute u'(t), u(t), v(t), and v'(t) into the quotient rule formula: \frac{0 \cdot (1 + 2^{-t}) - 40 \cdot (-2^{-t} \ln(2))}{(1 + 2^{-t})^2}. Simplify the expression to find the derivative of P(t).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Derivatives
A derivative represents the rate at which a function changes at any given point. It is a fundamental concept in calculus that measures how a function's output value changes as its input value changes. The derivative can be interpreted as the slope of the tangent line to the curve of the function at a specific point.
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Chain Rule
The chain rule is a formula for computing the derivative of a composite function. If a function is composed of two or more functions, the chain rule allows us to differentiate it by multiplying the derivative of the outer function by the derivative of the inner function. This is particularly useful when dealing with functions that include exponentials or other transformations.
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Exponential Functions
Exponential functions are mathematical functions of the form f(t) = a * b^t, where 'a' is a constant, 'b' is the base of the exponential, and 't' is the exponent. These functions are characterized by their rapid growth or decay and are commonly encountered in calculus. Understanding their properties is essential for differentiating functions that involve exponential terms.
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