9–61. Evaluate and simplify y'.

y = x^√x+1

Verified step by step guidance

First, identify the function given: \( y = x^{\sqrt{x} + 1} \). We need to find the derivative \( y' \).

To differentiate \( y = x^{\sqrt{x} + 1} \), recognize that it is a power function with a variable exponent. Use the logarithmic differentiation technique.

Take the natural logarithm of both sides: \( \ln(y) = \ln(x^{\sqrt{x} + 1}) \). This simplifies to \( \ln(y) = (\sqrt{x} + 1) \ln(x) \) using the logarithm power rule.

Differentiate both sides with respect to \( x \). For the left side, use the chain rule: \( \frac{d}{dx}[\ln(y)] = \frac{1}{y} \cdot y' \). For the right side, apply the product rule to \( (\sqrt{x} + 1) \ln(x) \).

After differentiating, solve for \( y' \) by multiplying both sides by \( y \). Substitute back \( y = x^{\sqrt{x} + 1} \) to express \( y' \) in terms of \( x \).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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. In this case, we need to apply differentiation rules to the function y = x^(√x + 1) to find y'.

Chain Rule

The Chain Rule is a technique used in differentiation when dealing with composite functions. It states that the derivative of a composite function is the derivative of the outer function multiplied by the derivative of the inner function. For the given function, recognizing the inner function (√x + 1) and the outer function (x raised to that power) is essential for correctly applying the Chain Rule.

Power Rule

The Power Rule is a basic rule in differentiation that states if f(x) = x^n, then f'(x) = n*x^(n-1). This rule simplifies the process of finding derivatives of polynomial and power functions. In the context of the given function, applying the Power Rule will be necessary after using the Chain Rule to differentiate the expression involving x raised to a variable exponent.

9–61. Evaluate and simplify y'.y = x^√x+1