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Magazine Chapter 3: Problem 51
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
Expert verified
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
01
Rewrite the function in terms of exponents
First, rewrite the function \(h(x)=x^{\sqrt{x}}\) using the property \(a^{\sqrt{b}}=a^{b^{\frac{1}{2}}}\):
$$h(x)=x^{x^{\frac{1}{2}}}$$
02
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 \(x\):
$$\frac{dh}{dx} = \frac{dg}{du}\cdot \frac{du}{dx}$$
03
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 \(u=x^{\frac{1}{2}}\):
$$\frac{dh}{dx}=\left(x^{\frac{1}{2}}x^{x^{\frac{1}{2}}-1}\right)\left(\frac{1}{2}x^{-\frac{1}{2}}\right)$$
04
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:
$$\frac{dh}{dx}= \frac{1}{2}x^{x^{\frac{1}{2}}-1}$$
05
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.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
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:
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.
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:
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.
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.
