Question

Derivatives of tower functions (or \(g^{h}\) ) Find the derivative of each function and evaluate the derivative at the given value of \(a\). $$h(x)=x^{\sqrt{x}} ; a=4$$

Short answer

The derivative of the function \(h(x)=x^{\sqrt{x}}\) is \(\frac{dh}{dx} = \frac{1}{2}x^{x^{\frac{1}{2}}-1}\). Its value at \(a=4\) is 2.

Step-by-step solution

Rewrite the function in terms of exponents

First, rewrite the function \(h(x)=x^{\sqrt{x}}\) using the property \(a^{\sqrt{b}}=a^{b^{\frac{1}{2}}}\): $$h(x)=x^{x^{\frac{1}{2}}}$$

Apply the chain rule

Apply the chain rule to the function \(h(x)=x^{x^{\frac{1}{2}}}\). We need to find the derivative of the outer function with respect to the inner function, then multiply by the derivative of the inner function. The outer function is \(g(u)=x^u\) and the inner function is \(u=x^{\frac{1}{2}}\). First, let's find the derivative of the outer function with respect to the inner function: $$\frac{dg}{du} = \frac{d}{du}(x^u) = ux^{u-1}$$ Next, find the derivative of the inner function: $$\frac{du}{dx} = \frac{d}{dx}(x^{\frac{1}{2}}) = \frac{1}{2}x^{-\frac{1}{2}}$$ By applying the chain rule, we can find the derivative of \(h(x)\) with respect to \(x\): $$\frac{dh}{dx} = \frac{dg}{du}\cdot \frac{du}{dx}$$

Multiply the derivatives

Multiply the derivatives we found in step 2 together: $$\frac{dh}{dx} = \left(ux^{u-1}\right)\left(\frac{1}{2}x^{-\frac{1}{2}}\right)$$ Substitute \(u=x^{\frac{1}{2}}\): $$\frac{dh}{dx}=\left(x^{\frac{1}{2}}x^{x^{\frac{1}{2}}-1}\right)\left(\frac{1}{2}x^{-\frac{1}{2}}\right)$$

Simplify the derivative

Simplify the expression for the derivative of \(h(x)\): $$\frac{dh}{dx}= \frac{1}{2}x^{-\frac{1}{2}}x^{\frac{1}{2}}x^{x^{\frac{1}{2}}-1}$$ Combine the exponents: $$\frac{dh}{dx}= \frac{1}{2}x^{x^{\frac{1}{2}}-1}$$

Evaluate the derivative at a = 4

Lastly, evaluate the derivative of \(h(x)\) at \(a=4\): $$\frac{dh}{dx}(4) = \frac{1}{2}4^{4^{\frac{1}{2}}-1}$$ Simplify: $$\frac{dh}{dx}(4) = \frac{1}{2}4^{2-1}$$ $$\frac{dh}{dx}(4) = \frac{1}{2}4^1$$ $$\frac{dh}{dx}(4) = 2$$ The derivative of the function \(h(x)=x^{\sqrt{x}}\) is \(\frac{dh}{dx} = \frac{1}{2}x^{x^{\frac{1}{2}}-1}\), and its value at \(a=4\) is 2.

Key concepts

Chain Rule

The Chain Rule is a fundamental concept in calculus used to find the derivative of a composite function. Composite functions are functions within functions, and the Chain Rule helps us break down the problem into simpler parts.

Here's how it works:

• Identify the outer function and the inner function. In the exercise, this was seen in how we expressed the problem as a function of another function, specifically, we treated \(h(x) = x^{\sqrt{x}}\) as having an outer function \(g(u) = x^u\) and an inner function \(u = x^{\frac{1}{2}}\).
• Take the derivative of the outer function with respect to the inner function. This involves basic power rules of differentiation.
• Find the derivative of the inner function with respect to \(x\).
• Multiply these derivatives together to get the final derivative of the composite function.

The Chain Rule is essential because it simplifies the differentiation of functions that would otherwise be difficult to handle. By breaking them into more manageable pieces, we can differentiate with ease, even for complex expressions.

Exponentiation

Exponentiation involves raising a number, known as the base, to the power of another number, called the exponent. In derivatives, functions involving exponents pose unique challenges, as seen in the exercise that dealt with \(h(x) = x^{\sqrt{x}}\).

A few key points about exponentiation:

• When differentiating exponentials, rules like the power rule \(\frac{d}{dx} (x^n) = nx^{n-1}\) are often utilized.
• In cases of variable exponents, you might need to use more complex rules like the Chain Rule, evident in differentiating \(x^{\sqrt{x}}\).
• Simplifying exponents by rewriting them in a more manageable form can make derivative calculations much smoother. In our problem, \(x^{\sqrt{x}}\) was transformed to \(x^{x^{\frac{1}{2}}}\) to simplify the process.

Understanding exponentiation thoroughly is vital as it connects directly with logarithmic differentiation for more challenging functions and enhances overall differentiation skills.

Differentiation Techniques

Differentiation is the mathematical process of finding the derivative, which reflects a function's rate of change. While there are several rules and techniques for differentiation, selecting the right technique depends on the given problem. Here are some crucial techniques:

• Power Rule: Applied when differentiating powers of \(x\). For \(x^n\), the derivative is \(nx^{n-1}\).
• Chain Rule: Used for composite functions, it involves taking the derivative of the outer function and multiplying it by the derivative of the inner function, just as seen in the exercise problem.
• Product and Quotient Rules: Essential when dealing with products or quotients of two functions. They are crucial when functions are multiplied or divided instead of just nested.

Different problems require the proper selection and application of these techniques. For the derivative of \(h(x) = x^{\sqrt{x}}\), it was essential to utilize the Chain Rule due to the composite structure, allowing us to find the exact rate of change for the function as needed.