Warm Up

Andrew Dickinson

Intro

Derivatives do not make up a significant portion of this course, however I think y’all could use some support on the chain rule. Use the chain rule on the following problems.

The chain rule

Sure, here’s a brief description and a quick example of the chain rule:

The chain rule is a fundamental derivative rule in calculus used to find the derivative of the composition of two or more functions. When one function is applied inside another, the chain rule allows us to differentiate the entire combination by taking the derivative of the outer function and multiplying it by the derivative of the inner function. The general formula is given by:

\[ \frac{d}{dx}(f(g(x))) = f'(g(x)) \cdot g'(x) \]

Ex.

Consider the function \(h(x) = (2x + 3)^3\). To find \(\frac{d}{dx}h(x)\), we let the inner function be \(g(x) = 2x + 3\) and the outer function be \(f(u) = u^3\). Applying the chain rule:

• The derivative of the outer function with respect to its argument \(u\) is \(f'(u) = 3u^2\).
• The derivative of the inner function with respect to \(x\) is \(g'(x) = 2\).

Using the chain rule:

\[ \frac{d}{dx}h(x) = f'(g(x)) \cdot g'(x) = 3(2x + 3)^2 \cdot 2 = 6(2x + 3)^2 \]

Questions

1. Let \(f(x) = (3x + 2)^2\). Compute \(\frac{d}{dx}f(x)\).

To compute \(\frac{d}{dx}f(x)\), apply the chain rule. Let \(u(x) = 3x + 2\) and \(f(u) = u^2\). Then, \(\frac{d}{du}f(u) = 2u\) and \(\frac{d}{dx}u(x) = 3\).

\[ \frac{d}{dx}f(x) = \frac{d}{du}f(u) \cdot \frac{d}{dx}u(x) = 2(3x + 2) \cdot 3 = 6(3x + 2) \]

2. Let \(g(x) = \sqrt{5x - 1}\). Compute \(\frac{d}{dx}g(x)\).

Apply the chain rule with \(v(x) = 5x - 1\) and \(g(v) = \sqrt{v}\). Then, \(\frac{d}{dv}g(v) = \frac{1}{2\sqrt{v}}\) and \(\frac{d}{dx}v(x) = 5\).

\[ \frac{d}{dx}g(x) = \frac{d}{dv}g(v) \cdot \frac{d}{dx}v(x) = \frac{1}{2\sqrt{5x - 1}} \cdot 5 = \frac{5}{2\sqrt{5x - 1}} \]

3. Let \(k(x) = \sum_{i=1}^{n} (Z_i - x W_i)^2\). Compute \(\frac{d}{dx}k(x)\).

The derivative of \(k(x)\) with respect to \(x\) involves the chain rule and the power rule.

\[ \begin{align*} \frac{d}{dx}k(x) &= \frac{d}{dx}\sum_{i=1}^{n} (Z_i - x W_i)^2 \\ &= \sum_{i=1}^{n} \frac{d}{dx}(Z_i - x W_i)^2 \\ &= \sum_{i=1}^{n} 2(Z_i - x W_i)(-W_i) \\ &= -2\sum_{i=1}^{n} W_i(Z_i - x W_i) \end{align*} \]