Question

By means of the definition of \(\$ 1\) differentiate each of the following functions: \(y=\frac{1}{x^{2}}\)

Short answer

Answer: The derivative of the function \(y=\frac{1}{x^{2}}\) with respect to x is \(\frac{dy}{dx} = -\frac{2}{x^{3}}\).

Step-by-step solution

Recall the definition of derivative \$1

The definition of derivative \$1 is given as: $$\frac{dy}{dx} = \lim_{h \to 0}\frac{f(x+h)-f(x)}{h}$$

Identify the function \(f(x)\)

From the given problem, \(y = \frac{1}{x^{2}}\), we identify the function as: $$f(x) = \frac{1}{x^{2}}$$

Find \(f(x+h)\)

For the given function \(f(x) = \frac{1}{x^{2}}\), we need to find \(f(x+h)\) by substituting \(x+h\) in place of x in the function: $$f(x+h) = \frac{1}{(x+h)^{2}}$$

Calculate \(f(x+h)-f(x)\)

Now, find the difference between \(f(x+h)\) and \(f(x)\): \begin{align*} f(x+h)-f(x) &= \frac{1}{(x+h)^{2}} - \frac{1}{x^{2}} \\ &= \frac{x^{2}-(x+h)^{2}}{x^{2}(x+h)^{2}} \end{align*}

Calculate the quotient \(\frac{f(x+h)-f(x)}{h}\)

Next, we need to calculate the quotient by dividing the difference \(f(x+h)-f(x)\) by \(h\): $$\frac{f(x+h)-f(x)}{h} = \frac{\frac{x^{2}-(x+h)^{2}}{x^{2}(x+h)^{2}}}{h}$$

Compute the limit

Apply the limit as h approaches 0: \begin{align*} \frac{dy}{dx} &= \lim_{h \to 0} \frac{\frac{x^{2}-(x+h)^{2}}{x^{2}(x+h)^{2}}}{h} \\ &= \lim_{h \to 0} \frac{x^{2}-(x+h)^{2}}{x^{2}(x+h)^{2} h} \\ &= \lim_{h \to 0} \frac{-2xh - h^{2}}{x^{2}(x+h)^{2} h} \end{align*}

Simplify the expression

Now, simplify the expression by factoring out an h from the numerator and then canceling it with the h in the denominator: \begin{align*} \frac{dy}{dx} &= \lim_{h \to 0} \frac{-h(2x + h)}{x^{2}(x+h)^{2} h} \\ &= \lim_{h \to 0} \frac{-(2x + h)}{x^{2}(x+h)^{2}} \end{align*}

Evaluate the limit

Finally, evaluate the limit as h approaches 0: \begin{align*} \frac{dy}{dx} &= \frac{-(2x + 0)}{x^{2}(x+0)^{2}} \\ &= \frac{-2x}{x^{4}} \\ &= -\frac{2}{x^{3}} \end{align*} So, the derivative of the function \(y=\frac{1}{x^{2}}\) with respect to \(x\) is $$\frac{dy}{dx} = -\frac{2}{x^{3}}.$$

Key concepts

Derivative Definition

Understanding the derivative of a function is fundamental to calculus. In its simplest terms, a derivative represents how a function changes as its input changes. Mathematically, it is the rate at which the function's output varies.

The formal definition involves the concept of a limit. For a function y = f(x), the derivative f'(x) or \(\frac{dy}{dx}\) at a point x is given by: \[\frac{dy}{dx} = \frac{f(x+h)-f(x)}{h}\] as \(h\) approaches zero. Here, \(h\) is an infinitesimally small number, signifying a slight change in the variable x. As h becomes very small, the ratio \(\frac{f(x+h)-f(x)}{h}\) gives the slope of the tangent to the curve y = f(x) at the point x.

By using the definition of the derivative, we can understand how a function's rate of change at a specific point is determined, and it provides the foundation for more advanced techniques in differentiating functions.

Limit Definition

A limit is a fundamental concept in calculus that describes the behavior of a function as its argument approaches a particular value. The limit of a function at a certain point gives us an idea of what value the function is approaching, even if it does not actually reach that value.

The limit of a function \(f(x)\) as \(x\) approaches a value \(a\) is represented as: \[ \text{lim}_{x \to a} f(x) \]

For the derivative process, we are interested in the limit as \(h\), the difference between two x-values, approaches zero. This allows us to find the instantaneous rate of change, or the derivative, of the function. In our specific example with the function \(y = \frac{1}{x^2}\), the limit is used to determine the exact value of the derivative as the value of \(h\) gets extremely close to zero.

Power Rule for Differentiation

The power rule is a basic yet powerful rule used to find the derivative of functions that are polynomials. When a function is written in the form \(f(x) = x^n\), where \(n\) is any real number, the power rule can be applied for differentiation.

According to the power rule, the derivative of \(f(x) = x^n\) is given by: \[f'(x) = nx^{n-1}\]

When we have a function with an inverse power, such as \(f(x) = \frac{1}{x^2}\) or \(f(x) = x^{-2}\), we can still apply the power rule. By rewriting the function as \(f(x) = x^{-2}\), we quickly find its derivative to be \(f'(x) = -2x^{-3}\), which simplifies to \(-\frac{2}{x^3}\). This demonstrates the utility of the power rule in finding derivatives quickly without going through the limit definition each time. However, using the definition of the derivative helps in understanding the underlying processes of differentiation.