Question

Evaluate the following derivatives. $$\frac{d}{d x}\left(2^{\left(x^{2}\right)}\right)$$

Short answer

Question: Find the derivative of the function \(f(x) = 2^{x^2}\) with respect to x. Answer: The derivative of the function \(f(x) = 2^{x^2}\) with respect to x is \(f'(x) = 2x(2^{x^2})\ln(2)\).

Step-by-step solution

Identify the inner and outer functions

The given function can be seen as a composition of two functions, an inner function and an outer function. The inner function is the exponent, which is \(x^2\), and the outer function is the exponential part, \(2^u\). Here, we will work with \(u\) as a temporary variable.

Apply the chain rule

To find the derivative of the composite function, we will apply the chain rule, which states that the derivative of a composite function is the product of the derivative of the outer function times the derivative of the inner function. In this case: $$\frac{d}{dx}(2^{x^{2}}) = \frac{d(2^u)}{du}\cdot\frac{du}{dx}$$

Find the derivative of the outer function

We will find the derivative of the outer function, \(2^u\), with respect to \(u\). The derivative of any exponential function with base \(a\) is given by: $$\frac{d(a^u)}{du}=a^u\ln(a)$$ Applying this rule to our outer function, we get: $$\frac{d(2^u)}{du}=2^u\ln(2)$$

Find the derivative of the inner function

The inner function is \(u=x^2\). We will find the derivative of this function with respect to \(x\). Since it is a simple power function, we can use the power rule for derivatives: $$\frac{du}{dx} = \frac{d(x^2)}{dx} = 2x$$

Substitute and multiply the derivatives to find the final result

Now that we have found the derivatives for both the inner and outer functions, we can substitute them back into our chain rule equation from Step 2: $$\frac{d}{dx}(2^{x^2})=\left(2^u\ln(2)\right)\cdot(2x)$$ Since \(u=x^2\), we can substitute it back into our expression: $$\frac{d}{dx}(2^{x^2})=\left(2^{x^2}\ln(2)\right)\cdot(2x)$$ The final result is: $$\frac{d}{dx}(2^{x^{2}}) = 2x(2^{x^2})\ln(2)$$

Key concepts

Chain Rule

When working with derivatives of composite functions, the chain rule is a powerful tool. The chain rule lets us differentiate functions that are nested within each other. We have a situation where one function is inside another, and we need to differentiate the outer function first, then multiply by the derivative of the inner function.
For example, consider a function \( f(g(x)) \). To differentiate it, the chain rule states:

• Differentiate the outer function, \( f(u) \), as if \( u = g(x) \)
• Differentiate the inner function \( g(x) \)
• Multiply these derivatives: \( f'(g(x)) \times g'(x) \)

Applying this process ensures we correctly handle composite functions. In our exercise, \( f(u) = 2^u \), and \( g(x) = x^2 \). Thus, the chain rule helps us calculate \( \frac{d}{dx}(2^{x^2}) \) by combining both derivatives step-by-step.

Exponential Functions

Exponential functions are fundamental in calculus, characterized by a constant base raised to a variable exponent. A general form is \( a^x \), where \( a \) is a constant, and \( x \) is the variable. Calculating derivatives with these involves a specific rule.
The derivative of an exponential function, \( a^u \), where \( u \) is any function of \( x \), is \( a^u \ln(a) \) times the derivative of \( u \). This results from the property of natural logarithms and the exponential base.
In the original solution, the function \( 2^{x^2} \) shows this vividly. Here, \( a = 2 \), and using the formula, \( \frac{d}{du}(2^u) = 2^u \ln(2) \), we apply the chain rule after finding the inner function's derivative for the full derivative solution.

Power Rule

The power rule is a basic yet critical tool in calculus, allowing us to differentiate power functions efficiently. These functions have the form \( x^n \), where \( n \) is any real number.
Using the power rule involves these steps:

• Bring down the exponent as a coefficient.
• Subtract one from the original exponent.

The formula for the power rule is \( \frac{d}{dx}(x^n) = nx^{n-1} \). This simple rule lets us swiftly compute derivatives of power functions.
In our exercise, the inner function \( x^2 \) applies the power rule directly. Differentiating gives \( \frac{d}{dx}(x^2) = 2x \), which is crucial for applying the chain rule to find the full derivative of the composite function \( 2^{x^2} \).