Understanding and Calculating the First Derivative of f(x) 2^{2x} - x^{21^{2/3}}

When dealing with calculus problems, understanding the concept of derivatives is crucial. In this article, we will explore the first derivative of the function f(x) 2^{2x} - x^{21^{2/3}} and break down the steps to find it.

Introduction to Derivatives

A derivative is a measure of how a function changes as its input changes. Specifically, the first derivative of a function at a given point gives the slope of the tangent line to the function at that point. This concept is fundamental in understanding the behavior of functions, including their maxima, minima, and points of inflection.

The Function in Question: f(x) 2^{2x} - x^{21^{2/3}}

The given function is a combination of two parts: a power of an exponential function 2^{2x} and a power of a polynomial function x^{21^{2/3}}. These functions might seem complex, but with the right tools, they can be differentiated with ease.

Breaking Down the Derivatives

Derivative of 2^{2x}

Let's start with the first term, 2^{2x}. To differentiate this, we will use the Chain Rule. The Chain Rule states that the derivative of a composition of functions g(h(x)) is the derivative of g evaluated at h(x) times the derivative of h(x).

2^{2x} can be rewritten as (2^2)^x 4^x. The derivative of 4^x is 4^x ln(4). Therefore, the derivative of 2^{2x} is:

[ frac{d}{dx}(2^{2x}) (2ln(2)) cdot 2^{2x} 2^{2x 1} ln(2) ]

Derivative of x^{21^{2/3}}

Now let's move on to the second term, x^{21^{2/3}}. The Chain Rule is also useful here. Consider x^{21^{2/3}} as a function of the form x^a, where a 21^{2/3}. The derivative of x^a is a cdot x^{a-1}. Therefore, the derivative of x^{21^{2/3}} is:

[ frac{d}{dx}(x^{21^{2/3}}) 21^{2/3} cdot x^{21^{2/3}-1} ]

Combining the Derivatives

To find the first derivative of the given function, we simply combine the derivatives of the two terms:

[ f'(x) 2^{2x 1} ln(2) - 21^{2/3} cdot x^{21^{2/3}-1} ]

Verification Using Wolfram Alpha

To verify our calculations, we can use the online computational engine Wolfram Alpha. By inputting the function 2^{2x} - x^{21^{2/3}} and asking for its derivative, Wolfram Alpha confirms our solution.

Conclusion

The first derivative of the function f(x) 2^{2x} - x^{21^{2/3}} is: