Introduction to Derivatives in Calculus

Calculus, a fundamental branch of mathematics, plays a crucial role in understanding various phenomena ranging from physics and engineering to economics and data analysis. One of the core concepts in calculus is the derivative, which helps us understand the rate of change of functions. This article focuses on finding the derivative of the function (y x^{x^{1/2}}), a more complex problem involving exponential and logarithmic functions. By employing techniques such as logarithmic differentiation, we can break down this problem into more manageable parts.

Understanding (f^{g^h})

Let's first clarify the notation and its implications: (f^{g^h}) is distinct from (f^{g^h} eq f^g^h f^{gh}). Understanding this is crucial for the correct differentiation of expressions. For instance, in the given function, we have (x^{x^{1/2}}).

Method 1: Using Logarithmic Differentiation

To find the derivative of (y x^{x^{1/2}}), we can use logarithmic differentiation. Taking the natural logarithm of both sides of the equation, we obtain:

(ln y lnleft(x^{sqrt{x}}right)) (ln y sqrt{x} ln x)

Next, we differentiate both sides with respect to (x):

(frac{1}{y} frac{dy}{dx} frac{1}{2sqrt{x}} ln x frac{sqrt{x}}{x} frac{1}{sqrt{x}} ln sqrt{x})

Multiplying both sides by (y), we get:

(frac{dy}{dx} y cdot frac{1}{sqrt{x}} ln sqrt{x} x^{sqrt{x}} cdot frac{1}{sqrt{x}} ln sqrt{x}) (frac{dy}{dx} x^{sqrt{x} - frac{1}{2}} ln sqrt{x})

Method 2: Using Complex Logarithmic Manipulation

Alternatively, another approach involves a more complex logarithmic manipulation. Consider the function (y x^{x^{1/2}}). We take the natural logarithm of both sides:

(ln y lnleft(x^{sqrt{x}}right) sqrt{x} ln x)

Next, we derive both sides with respect to (x), employing the product rule:

(frac{1}{y} frac{dy}{dx} frac{1}{2sqrt{x}} ln x frac{sqrt{x}}{x})

Multiplying both sides by (y), we get:

(frac{dy}{dx} y left( frac{1}{2sqrt{x}} ln x frac{sqrt{x}}{x} right) x^{sqrt{x}} left( frac{1}{2sqrt{x}} ln x frac{sqrt{x}}{x} right)) (frac{dy}{dx} x^{sqrt{x}-frac{1}{2}} left( frac{1}{2}ln x frac{sqrt{x}}{x} right))

After simplifying, we obtain:

(frac{dy}{dx} frac{x^{sqrt{x}-frac{1}{2}}}{sqrt{x}} ln x)

Applying Exponential Functions

Given the expression (f(x) x^{sqrt{x}} e^{sqrt{x}ln x}), we can find its derivative by recognizing that:

(f'(x) e^{sqrt{x}ln x} cdot frac{sqrt{x}}{x} frac{ln x}{2sqrt{x}} frac{e^{sqrt{x}ln x}}{sqrt{x}}left(frac{1}{2} ln xright))

Similarly, for the function (f(x) x^{x^{1/2}} e^{xln x/2}), we find:

(f'(x) frac{1}{2} e^{xln x/2} ln x)

Conclusion

Understanding how to find the derivative of complex functions like (x^{x^{1/2}}) is crucial in various fields of study. By leveraging logarithmic differentiation and exponential functions, we can break down such problems into simpler, more manageable terms. This method not only provides a deeper insight into the function but also showcases the powerful techniques of calculus.