Textbook Question
7–28. Derivatives Evaluate the following derivatives.
d/dx ((2x)⁴ˣ)
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6. Derivatives of Inverse, Exponential, & Logarithmic Functions
Derivatives of Exponential & Logarithmic Functions
4:23 minutesProblem 7.1.20
Textbook Question
7–28. Derivatives Evaluate the following derivatives.
d/dt (t^{1/t})
Verified step by step guidance1
Step 1: Recognize that the function t^(1/t) involves both a variable base and a variable exponent. To differentiate it, use logarithmic differentiation. Start by taking the natural logarithm of both sides: let y = t^(1/t), then ln(y) = (1/t) * ln(t).
Step 2: Differentiate both sides of the equation with respect to t. For the left-hand side, use implicit differentiation: d/dt[ln(y)] = (1/y) * dy/dt. For the right-hand side, apply the product rule to differentiate (1/t) * ln(t).
Step 3: Apply the product rule to (1/t) * ln(t). The product rule states that d/dt[u*v] = u'v + uv'. Here, u = 1/t and v = ln(t). Differentiate u and v separately: u' = -1/t^2 and v' = 1/t.
Step 4: Substitute the derivatives into the product rule formula: d/dt[(1/t) * ln(t)] = (-1/t^2) * ln(t) + (1/t) * (1/t). Simplify the expression to combine terms.
Step 5: Solve for dy/dt by multiplying through by y (recall y = t^(1/t)) to isolate dy/dt. Substitute back y = t^(1/t) into the final expression to express the derivative in terms of 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 of change of a function with respect to a variable. It is a fundamental concept in calculus that allows us to understand how a function behaves locally. The derivative of a function at a point gives the slope of the tangent line to the function's graph at that point, indicating how the function's output changes as the input changes.
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Chain Rule
The chain rule is a formula for computing the derivative of the composition of two or more functions. It states that if you have a function that is composed of another function, the derivative can be found by multiplying the derivative of the outer function by the derivative of the inner function. This is particularly useful when dealing with functions that involve exponentiation or logarithms, as seen in the given expression.
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Exponential Functions
Exponential functions are functions of the form f(t) = a^g(t), where a is a constant and g(t) is a function of t. In the context of the problem, t^{1/t} can be rewritten using properties of exponents, which can simplify the differentiation process. Understanding how to manipulate and differentiate exponential functions is crucial for solving problems involving derivatives of such forms.
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