Textbook Question
22–36. Derivatives Find the derivatives of the following functions.
f(x) = tanh²x
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6. Derivatives of Inverse, Exponential, & Logarithmic Functions
Derivatives of Exponential & Logarithmic Functions
3:18 minutesProblem 7.3.32
Textbook Question
22–36. Derivatives Find the derivatives of the following functions.
f(t) = 2 tanh⁻¹ √t
Verified step by step guidance1
Step 1: Recognize that the function f(t) = 2 tanh⁻¹(√t) involves the inverse hyperbolic tangent function (tanh⁻¹) and a square root. To differentiate this, we will use the chain rule and the derivative formula for tanh⁻¹(x).
Step 2: Recall the derivative formula for tanh⁻¹(x): d/dx[tanh⁻¹(x)] = 1 / (1 - x²). This will be applied to the argument √t.
Step 3: Apply the chain rule. First, differentiate the outer function 2 tanh⁻¹(√t) with respect to √t, which gives 2 * (1 / (1 - (√t)²)).
Step 4: Next, differentiate the inner function √t with respect to t. Recall that d/dt[√t] = 1 / (2√t). Multiply this result with the derivative from Step 3.
Step 5: Combine the results from Steps 3 and 4 to express the derivative of f(t). The final derivative will be f'(t) = 2 * (1 / (1 - t)) * (1 / (2√t)). Simplify if needed.
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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 provides information about the slope of the tangent line to the function's graph at any given point. The process of finding a derivative is called differentiation, and it is essential for understanding how functions behave and change.
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Inverse Hyperbolic Functions
The function tanh⁻¹, or inverse hyperbolic tangent, is the inverse of the hyperbolic tangent function. It is used to find the value of the original variable when given a hyperbolic tangent value. Understanding how to differentiate inverse hyperbolic functions is crucial for solving problems involving these functions, as they have specific derivative formulas.
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Chain Rule
The chain rule is a fundamental technique in calculus used to differentiate composite functions. It states that if a function is composed of two or more functions, the derivative can be found by multiplying the derivative of the outer function by the derivative of the inner function. This rule is particularly important when dealing with functions like f(t) = 2 tanh⁻¹(√t), where the argument of the inverse function is itself a function of t.
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