### Lecture on the Quotient Rule for Derivatives Today, we'll explore the quotient rule for derivatives, an indispensable tool in calculus for finding the derivative of the quotient of two functions. For a more illustrative example, let's consider the derivative of the quotient $\frac{x^3}{x}$. We'll approach this in two ways: first, by simplifying the expression and then using the power rule, and second, by directly applying the quotient rule. --- #### Simplifying Before Differentiation Let's begin by simplifying $\frac{x^3}{x}$. According to the laws of exponents, when we divide two expressions with the same base, we subtract their exponents: $$ \frac{x^3}{x} = x^{3-1} = x^2 $$ Now, to find the derivative of $x^2$, we use the power rule. The power rule states that if $f(x) = x^n$, then $f'(x) = nx^{n-1}$. Applying this to $x^2$, we get: $$ \frac{d}{dx}x^2 = 2x^{2-1} = 2x $$ Thus, by simplifying the expression first and then differentiating, we find that the derivative of $\frac{x^3}{x}$ is $2x$. --- #### Using the Quotient Rule Now, let's approach the same problem using the quotient rule. The quotient rule states that if we have two functions $u(x)$ and $v(x)$, and $y = \frac{u(x)}{v(x)}$, then the derivative of $y$ with respect to $x$ is given by: $$ y' = \frac{u'v - uv'}{v^2} $$ --- In our case, $u(x) = x^3$ and $v(x) = x$. To apply the quotient rule, we first need to find the derivatives of $u(x)$ and $v(x)$. Using the power rule: - The derivative of $u(x) = x^3$ is $u'(x) = 3x^{3-1} = 3x^2$. - The derivative of $v(x) = x$ is $v'(x) = 1x^{1-1} = 1$ --- Housing the information in this table as follows: | | | |--- | ---| | $u=x^3$ | $v=x$ | | $u'=3x^2$ | $v'=1$ | --- Applying the quotient rule: \begin{align*} y' &= \frac{u'v - uv'}{v^2} \\ &= \frac{(3x^2)(x) - (x^3)(1)}{x^2} \\ &= \frac{3x^3 - x^3}{x^2} \\ &= \frac{2x^3}{x^2} \\ &= 2x \end{align*} --- Interestingly, both methods yield the same result: $2x$. This not only demonstrates the consistency and reliability of calculus principles but also shows two different approaches to solving the same problem. --- #### Conclusion By comparing these two methods, we can see the flexibility in choosing an approach based on the problem at hand. Simplifying before differentiating can often make the process quicker and more straightforward. However, the quotient rule is indispensable when dealing with more complex functions that cannot be easily simplified or when the functions represent quantities that need to be divided. This highlights the power and utility of calculus tools in mathematical analysis and problem-solving.