Problem 7 - Single Variable Calculus

Question

Find the differential of the function at the indicated number. $$ f(x)=2 x^{1 / 4}+3 x^{-1 / 2} ; \quad x=1 $$

Step-by-Step Solution

Answer

The differential of the function \(f(x) = 2x^{1/4} + 3x^{-1/2}\) at \(x = 1\) is \(-1\).

Step 1: Identify the function

We are given the function \(f(x) = 2x^{1/4} + 3x^{-1/2}\).

Step 2: Differentiate the function with respect to x

To differentiate the function, we need to apply the power rule, which states that \(\frac{d}{dx}(ax^n) = nax^{(n-1)}\): \( \frac{d}{dx}(2x^{1/4}) = 2*(1/4)x^{(1/4 - 1)} = \frac{1}{2}x^{-3/4} \) \( \frac{d}{dx}(3x^{-1/2}) = -\frac{3}{2}x^{(-1/2 - 1)} = -\frac{3}{2}x^{-3/2} \) Now, add the derivatives to get the overall derivative of the function: \( f'(x) = \frac{1}{2}x^{-3/4} - \frac{3}{2}x^{-3/2} \)

Step 3: Evaluate the derivative at the indicated point

Now, we need to evaluate the derivative at the point \(x = 1\): \( f'(1) = \frac{1}{2}(1)^{-3/4} - \frac{3}{2}(1)^{-3/2} = \frac{1}{2} - \frac{3}{2} = -1 \) So, the differential of the function at \(x = 1\) is \(-1\).

Key Concepts

Differentiation

Differentiation is a fundamental concept in calculus that involves finding the derivative of a function. This process gives us the rate at which a function's output changes with respect to changes in its input. Essentially, it tells us how sensitive the function is to small changes in the input variable.

When we find the derivative of a function, we are computing a new function that can predict the slope of the tangent line to the original function at any given point. This is incredibly useful in physics for finding velocities and accelerations, in economics for finding marginal costs or revenues, and in other fields where change rates are essential.

Power Rule

The power rule is a quick and straightforward differentiation technique that applies to functions where the variable has an exponent. According to the power rule, when you have a function of the form \(a x^n\), the derivative of that function is given by \(n a x^{(n-1)}\).

This rule greatly simplifies the process of finding the derivative of polynomial functions. For example, if you have the function \(f(x) = x^3\), applying the power rule would give you its derivative as \(f'(x) = 3x^2\). The power rule can also be extended to work with negative and fractional exponents, allowing us to handle a wide variety of functions with ease.

Derivative Evaluation

After finding a function's derivative, the next step is usually to evaluate the derivative at a specific point, which gives the instantaneous rate of change of the function at that point. The evaluation involves plugging the value of the input into the derivative formula.

For instance, if we have a derivative function \(f'(x)\), and we need to find the rate of change at \(x = c\), we substitute \(c\) for \(x\) in the derivative, resulting in \(f'(c)\). The numerical value of \(f'(c)\) tells us how fast the function is changing at \(x = c\), and can indicate whether the function is increasing or decreasing at that point.

Calculus

Calculus is a branch of mathematics that studies how things change. It provides a framework for modeling systems in which there is change, and a way to deduce the predictions of such models.

There are two major branches of calculus: differential calculus (concerned with the concept of a derivative) and integral calculus (concerned with the concept of an integral). Together, these two branches are fundamental to many fields of science and engineering, enabling professionals to predict and model complex systems. By understanding rates of change (differential calculus) and accumulation of quantities (integral calculus), one can solve problems related to motion, electricity, heat, light, harmonics, acoustics, astronomy, and even quantum mechanics.