Problem 35 - Calculus Early Transcendentals

Question

Derivatives Evaluate the derivatives of the following functions. \(h(x)=2^{\left(x^{2}\right)}\)

Step-by-Step Solution

Answer

Answer: The derivative is \(h'(x) = 2x \cdot 2^{x^2} \ln(2)\).

Step 1: Identify the Inner and Outer Functions

In this case, the inner function is \(u(x) = x^2\), and the outer function is \(g(u) = 2^u\) (where \(u\) is a function of \(x\)).

Step 2: Find the Derivative of the Inner Function

We need to find the derivative of the inner function, which is \(u'(x) = \frac{d}{dx}(x^2)\). Using the power rule, we obtain $$u'(x) = 2x$$

Step 3: Find the Derivative of the Outer Function

Now we need to find the derivative of the outer function with respect to \(u\). Using the power rule, we get $$g'(u) = \frac{d}{du}(2^u) = 2^u\ln(2)$$

Step 4: Apply the Chain Rule

According to the chain rule, we should multiply the derivative of the outer function evaluated at the inner function with the derivative of the inner function: $$h'(x) = g'(u(x)) \cdot u'(x) = g'(x^2) \cdot 2x$$

Step 5: Substitute and Simplify

Now, we can substitute the expression we found for \(g'(u)\) into the equation and simplify: $$h'(x) = 2^{x^2}\ln(2) \cdot 2x$$ The derivative of the function \(h(x) = 2^{x^2}\) is: $$h'(x) = 2x\cdot2^{x^2}\ln(2)$$

Key Concepts

Chain Rule

Understanding the chain rule is a critical part of calculus, especially when dealing with composite functions. Think of a composite function like a Russian doll, where one function is nested inside another. To differentiate these functions, you can't treat them as separate entities; they influence one another. The chain rule is the tool that helps us tackle this problem. It tells us to first differentiate the outer function (the big doll) with respect to the inner function (the smaller doll inside), and then multiply that by the derivative of the inner function with respect to the original variable.

For the exercise \(h(x)=2^{(x^{2})}\), the chain rule is used in Step 4. First, the outer function which is \(2^u\) (with \(u = x^2\)) is differentiated with respect to its own variable \(u\), and then this derivative is multiplied by the derivative of the inner function \(u'(x) = 2x\). This process effectively 'chains' the derivatives together, allowing us to find the derivative of the entire composite function.

Power Rule

When facing functions that are expressed as a variable raised to a power, the power rule is your best friend. This rule states that to find the derivative of \(x^n\), you can bring down the exponent \(n\), multiply it by the base \(x\), and then subtract one from the exponent. Formally, it's written as \(\frac{d}{dx}(x^n) = nx^{n-1}\).

In the given exercise, the power rule is applied in Step 2 to differentiate the inner function \(u(x) = x^2\). Here, the power rule simplifies the differentiation process considerably since the exponent is simply a 2, which yields \(u'(x) = 2x\). Not only does the power rule apply to polynomials, but it's also an integral part of finding derivatives in many other mathematical scenarios.

Exponential Functions Derivatives

Exponential functions, such as \(2^x\), have their own pattern of differentiation. Unlike polynomials that follow the power rule, exponential functions with a constant base and a variable exponent require a different approach. The derivative of \(a^x\) with respect to \(x\) is \(a^x\ln(a)\), where \(a\) is a constant and \(\ln\) denotes the natural logarithm.

In Step 3 of the exercise, we apply this principle to find the derivative of the outer function \(g(u) = 2^u\). It is crucial to remember here that the base \(2\) is a constant. After differentiating, we obtain \(g'(u) = 2^u\ln(2)\), which is then used in conjunction with the chain rule to find the overall derivative. Exponential functions' derivatives frequently appear in growth and decay problems, making them a vital concept in both mathematics and applied sciences.