{Use of Tech} A family of superexponential functions Let ƒ(x) = (a + x)ˣ , where a \u003e 0.c. Compute ƒ'. Then graphƒ and ƒ' for a = 0.5, 1, 2, and 3.

Ch. 4 - Applications of the Derivative

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Ch. 4 - Applications of the Derivative

문제 4.R.118c

4장, 문제 4.R.118c

{Use of Tech} A family of superexponential functions Let ƒ(x) = (a + x)ˣ , where a > 0.

c. Compute ƒ'. Then graphƒ and ƒ' for a = 0.5, 1, 2, and 3.

검증된 단계별 안내

To find the derivative ƒ'(x) of the function ƒ(x) = (a + x)^x, we will use logarithmic differentiation. Start by taking the natural logarithm of both sides: ln(ƒ(x)) = ln((a + x)^x).

Apply the logarithmic identity ln((a + x)^x) = x * ln(a + x). This simplifies the expression to ln(ƒ(x)) = x * ln(a + x).

Differentiate both sides with respect to x. For the left side, use the chain rule: d/dx[ln(ƒ(x))] = ƒ'(x)/ƒ(x). For the right side, use the product rule: d/dx[x * ln(a + x)] = 1 * ln(a + x) + x * (1/(a + x)).

Combine the derivatives: ƒ'(x)/ƒ(x) = ln(a + x) + x/(a + x). Solve for ƒ'(x) by multiplying both sides by ƒ(x): ƒ'(x) = ƒ(x) * (ln(a + x) + x/(a + x)).

Substitute ƒ(x) = (a + x)^x back into the expression for ƒ'(x): ƒ'(x) = (a + x)^x * (ln(a + x) + x/(a + x)). Now, you can graph ƒ(x) and ƒ'(x) for the given values of a: 0.5, 1, 2, and 3.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Differentiation

Differentiation is a fundamental concept in calculus that involves finding the derivative of a function. The derivative represents the rate of change of the function with respect to its variable. For the function ƒ(x) = (a + x)ˣ, applying the rules of differentiation, particularly the product and chain rules, is essential to compute ƒ'.

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 given function ƒ(x) = (a + x)ˣ, the base of the exponent is not constant, which introduces complexity in its behavior. Understanding the properties of exponential growth and how they affect the graph is crucial for analyzing ƒ and its derivative.

Graphing Functions

Graphing functions involves plotting the values of a function and its derivative on a coordinate plane to visualize their behavior. For the function ƒ(x) and its derivative ƒ', it is important to consider how changes in the parameter 'a' affect the shape and position of the graphs. This visual representation aids in understanding the relationship between the function and its rate of change.

{Use of Tech} A family of superexponential functions Let ƒ(x) = (a + x)ˣ , where a > 0.c. Compute ƒ'. Then graphƒ and ƒ' for a = 0.5, 1, 2, and 3.