Derivative Of 3 Square Root Of X

Understanding the Derivative of the Cube Root of x

The derivative of a function describes its instantaneous rate of change at any given point. Understanding derivatives is fundamental to calculus and has widespread applications in various fields, from physics and engineering to economics and finance. This article delves into the intricacies of finding the derivative of the cube root of x, exploring various approaches and highlighting crucial concepts along the way.

Defining the Function

Before we embark on finding the derivative, let's precisely define the function we're working with. The cube root of x is mathematically represented as:

f(x) = ³√x

This can also be written using exponential notation as:

f(x) = x^(1/3)

This latter representation is particularly useful when applying the power rule of differentiation.

Method 1: Using the Power Rule

The power rule is a fundamental theorem in differential calculus that simplifies the process of finding the derivative of functions in the form xn. The rule states that the derivative of xn is nx(n-1).

Applying the power rule to our function, f(x) = x^(1/3), we get:

f'(x) = (1/3)x^((1/3) - 1)

Simplifying the exponent:

f'(x) = (1/3)x^(-2/3)

This can be rewritten in radical form as:

f'(x) = 1 / (3x^(2/3))

f'(x) = 1 / (3³√x²)

Therefore, the derivative of the cube root of x is 1 / (3x^(2/3)) or 1 / (3³√x²). This result is valid for all x > 0.

Understanding the Result

The derivative, f'(x) = 1 / (3x^(2/3)), tells us the instantaneous rate of change of the cube root function at any point x. Observe that the derivative is always positive for x > 0, indicating that the cube root function is always increasing in this domain. Furthermore, the derivative approaches infinity as x approaches 0, reflecting the steepness of the cube root function near the origin. As x increases, the derivative decreases, reflecting the diminishing rate of increase of the cube root function.

Method 2: Using the Definition of the Derivative

The derivative of a function can also be defined using limits. This approach provides a more fundamental understanding of the concept of a derivative. The definition of the derivative is:

f'(x) = lim (h→0) [(f(x + h) - f(x)) / h]

Let's apply this definition to our function, f(x) = x^(1/3):

f'(x) = lim (h→0) [((x + h)^(1/3) - x^(1/3)) / h]

This limit is not straightforward to evaluate directly. We can employ a technique known as rationalizing the numerator. To do this, we use the difference of cubes factorization:

a³ - b³ = (a - b)(a² + ab + b²)

In our case, a = (x + h)^(1/3) and b = x^(1/3). We multiply the expression by the conjugate:

[(x + h)^(1/3) + x^(1/3)] / [(x + h)^(1/3) + x^(1/3)]

After simplification, we obtain:

f'(x) = lim (h→0) [((x + h) - x) / (h[(x + h)^(2/3) + (x + h)^(1/3)x^(1/3) + x^(2/3)])]

f'(x) = lim (h→0) [h / (h[(x + h)^(2/3) + (x + h)^(1/3)x^(1/3) + x^(2/3)])]

The 'h' in the numerator and denominator cancels out, leaving:

f'(x) = lim (h→0) [1 / ((x + h)^(2/3) + (x + h)^(1/3)x^(1/3) + x^(2/3))]

As h approaches 0, we get:

f'(x) = 1 / (x^(2/3) + x^(1/3)x^(1/3) + x^(2/3))

f'(x) = 1 / (3x^(2/3))

This is the same result as obtained using the power rule.

Method 3: Implicit Differentiation (for a related problem)

While not directly applicable to f(x) = x^(1/3), implicit differentiation is a powerful technique for finding derivatives when dealing with implicit functions. Consider the related problem of finding the derivative of y, where y³ = x. Here, y is implicitly defined as a function of x.

Differentiating both sides with respect to x, we get:

3y² (dy/dx) = 1

Solving for dy/dx:

dy/dx = 1 / (3y²)

Since y = x^(1/3), we can substitute this back into the equation:

dy/dx = 1 / (3(x^(1/3))²) = 1 / (3x^(2/3))

This again yields the same derivative as before, confirming our previous results.

Applications of the Derivative

The derivative of the cube root function, 1 / (3x^(2/3)), has many applications in various fields:

Optimization Problems

In optimization problems, finding the maximum or minimum values of a function is crucial. The derivative helps identify critical points where the function's slope is zero, indicating potential maxima or minima.

Physics

In physics, the derivative represents instantaneous velocity (the derivative of displacement with respect to time) and acceleration (the derivative of velocity with respect to time). The cube root function, while perhaps less common than other functions in basic physics, can appear in models involving, for example, certain types of diffusion processes.

Economics

In economics, the derivative can represent marginal cost, marginal revenue, or marginal profit. These concepts are essential for understanding economic decision-making.

Higher-Order Derivatives

It's also possible to compute higher-order derivatives. The second derivative, denoted f''(x), represents the rate of change of the first derivative. For our function, the second derivative is obtained by differentiating f'(x) = (1/3)x^(-2/3):

f''(x) = (1/3) * (-2/3) * x^(-5/3) = -2 / (9x^(5/3))

Similarly, higher-order derivatives can be calculated by repeatedly differentiating the previous derivative.

Conclusion

Finding the derivative of the cube root of x, whether using the power rule, the definition of the derivative, or implicit differentiation (for a related problem), consistently yields the result 1 / (3x^(2/3)) for x>0. Understanding this derivative is crucial for solving a wide range of problems in calculus and its various applications. This article has explored multiple approaches, emphasizing the conceptual understanding alongside the mathematical calculations, equipping you with a comprehensive grasp of this important concept. Remember to always consider the domain of the function and its derivative when working with these mathematical tools.