Textbook Question
22–36. Derivatives Find the derivatives of the following functions.
f(t) = 2 tanh⁻¹ √t
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6. Derivatives of Inverse, Exponential, & Logarithmic Functions
Derivatives of Exponential & Logarithmic Functions
2:27 minutesProblem 7.1.11
Textbook Question
7–28. Derivatives Evaluate the following derivatives.
d/dx ((ln 2x)⁻⁵)
Verified step by step guidance1
Step 1: Recognize that the given function is ((ln(2x))⁻⁵). This is a composite function, so we will need to use the chain rule to differentiate it.
Step 2: Apply the chain rule. Let u = ln(2x), so the function becomes u⁻⁵. The derivative of u⁻⁵ with respect to u is -5u⁻⁶.
Step 3: Differentiate u = ln(2x) with respect to x. Using the chain rule again, the derivative of ln(2x) is (1/(2x)) * 2, which simplifies to 1/x.
Step 4: Combine the results from Step 2 and Step 3. Multiply the derivative of u⁻⁵ (-5u⁻⁶) by the derivative of u (1/x). Substitute u = ln(2x) back into the expression.
Step 5: The final derivative is -5(ln(2x))⁻⁶ * (1/x). This is the simplified form of the derivative.
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Video duration:
2mKey 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 a variable. It is a fundamental concept in calculus that allows us to determine how a function behaves as its input changes. The notation d/dx indicates differentiation with respect to the variable x, and derivatives can be computed using various rules, such as the power rule, product rule, and chain rule.
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Derivatives
Chain Rule
The chain rule is a formula for computing the derivative of a composite function. If a function y is defined as a function of u, which in turn is a function of x (y = f(u) and u = g(x)), the chain rule states that the derivative dy/dx is the product of the derivative of f with respect to u and the derivative of g with respect to x. This is essential for differentiating functions like (ln(2x))⁻⁵, where the inner function is ln(2x).
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Natural Logarithm
The natural logarithm, denoted as ln, is the logarithm to the base e, where e is approximately 2.71828. It is an important function in calculus, particularly in differentiation and integration. The derivative of ln(x) is 1/x, and when dealing with expressions involving ln, understanding its properties and how to differentiate it is crucial for solving problems involving logarithmic functions.
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