Multiple Choice
Determine the intervals for which the function is concave up or concave down. State the inflection points.
6. Derivatives of Inverse, Exponential, & Logarithmic Functions
Derivatives of Exponential & Logarithmic Functions
2:18 minutesProblem 3.9.48
Textbook Question
15–48. Derivatives Find the derivative of the following functions.
s(t) = cos 2^t
Verified step by step guidance1
Identify the function s(t) = cos(2^t). This is a composition of functions, where the outer function is cos(u) and the inner function is u = 2^t.
Apply the chain rule for differentiation, which states that the derivative of a composite function f(g(x)) is f'(g(x)) * g'(x).
Differentiate the outer function with respect to its argument: the derivative of cos(u) is -sin(u).
Differentiate the inner function u = 2^t with respect to t. This requires using the exponential rule: the derivative of a^x is a^x * ln(a). Therefore, the derivative of 2^t is 2^t * ln(2).
Combine the results using the chain rule: the derivative of s(t) is -sin(2^t) * (2^t * ln(2)).
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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 of change of a function with respect to its variable. It is a fundamental concept in calculus that allows us to determine how a function behaves at any given point. The derivative is often denoted as f'(x) or dy/dx, and it 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 y = f(g(x)) is composed of two functions, the chain rule states that the derivative is found by multiplying the derivative of the outer function f with the derivative of the inner function g. This rule is essential when differentiating functions that involve other functions, such as exponential or trigonometric functions.
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Exponential Functions
Exponential functions are mathematical functions of the form f(x) = a^x, where 'a' is a constant and 'x' is the variable. In the context of derivatives, the derivative of an exponential function can be computed using the natural logarithm and the chain rule. Understanding how to differentiate exponential functions is crucial for solving problems involving growth and decay, as well as in various applications across science and engineering.
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