The Chain Rule

Calculate the derivative of the following composite polynomial functions, simplifying the result as much as possible:

1. a) $f(x) = (1 - x)^5$
2. b) $g(x) = (2x + 3)^2$

Given the function $f(x) = (3x^2 - 5)^4$, identify the "outer" function and the "inner" function. Once identified, apply the chain rule to find its derivative $f'(x)$.

Observe the following diagram representing the composition of two functions $u(x)$ and $v(x)$. If we know that $f(x) = (v \circ u)(x)$, use the information in the diagram to calculate $f'(x)$.

$u(x) = x^3 + 1 \quad \longrightarrow \quad v(u) = \sqrt{u}$

Calculate the derivative of the following composite polynomial function and find the gradient of the tangent line to its graph at the point with abscissa $x = 1$:

$f(x) = (x^3 - 2x^2 + 3)^5$

Calculate the derivative of the following radical function:

$g(x) = \sqrt{5x + 2}$

Use the structure of the chain rule for powers, remembering that $\sqrt{u} = u^{1/2}$.

Find the derivative of $h(x) = \ln(x^2 + 3x)$. To do so, remember that if $y = \ln(u)$, then $y' = \frac{u'}{u}$.

Given the composite exponential function $g(x) = e^{x^2 - 4x + 1}$, determine the points on the curve where the tangent line is horizontal.

Determine the value of the constant parameter $k$ in the function $f(x) = \ln(kx^2 + 1)$ knowing that the slope of the tangent line to the graph of the function at the point with abscissa $x = 1$ is equal to $1$.

The chain rule tells us that if we have a composition of functions $y = f(g(x))$, its derivative is $y' = f'(g(x)) \times g'(x)$.

Apply this concept to differentiate the trigonometric function:

$f(x) = \sin(4x - 1)$

Given the function $h(x) = \log_2 \left( \sqrt[3]{x^3 + 3x^2} \right)$, calculate the value of its derivative at the point with abscissa $x = 1$. Note: Remember to apply the chain rule for logarithms in bases other than $e$.

Calculate the derivative of the following expression, simplifying the result as much as possible using trigonometric identities or properties of logarithms:

In the marketing department of a fashion startup, they have estimated that the number of registered users on their website follows the function:

$U(t) = e^{0.5t + 2}$

where $t$ is the time in months. Find the expression for the instantaneous rate of growth of the users (that is, the derivative $U'(t)$).

An electronics manufacturer determines that the total cost (in pesos) of producing $x$ units of a new sensor is given by:

$C(x) = (x + 10)^{3}$

Calculate the marginal cost when $x$ units are produced (that is, find $C'(x)$).

An electronic component manufacturing company has estimated that the total cost (in thousands of pesos) of producing $x$ units of a new processor model is given by the function:

$C(x) = \sqrt{2x^2 + 5x + 100}$

Obtain the expression for the marginal cost $C'(x)$ using the chain rule and calculate this cost when exactly $10$ units are produced.

The number of followers of a well-known influencer on a social network, measured in thousands of people, follows a growth model given by the function:

$S(t) = 50 \times \ln(t^2 + 4t + 5)$

where $t$ is the time in months since the launch of her advertising campaign. The graph shows the function and the point corresponding to $t = 3$.

Calculate the growth rate of followers (the derivative) in the third month ($t = 3$).

A digital marketing agency has determined that the number of active users on a social platform, $U(t)$, measured in thousands, follows a function depending on advertising spend $x$ (in thousands of pesos), which in turn varies over time $t$ (in months). The functions are:

Calculate the instantaneous rate of change of the number of users with respect to time when exactly $t = 2$ months have elapsed. Interpret whether user growth is accelerating or slowing down at that moment.

Find the derivative of the following function, which combines the quotient rule with the chain rule:

$h(x) = \frac{(3x - 1)^2}{x^2 + 1}$

Simplify the final result as much as possible.

The market value of a technology share (in pesos) has fluctuated during an 8-hour trading session according to the function:

$V(t) = 200 + 40 \times \sqrt{(t - 4)^2 + 9}$

where $t$ is the time in hours ($0 \le t \le 8$). Determine the instantaneous rate of change of the share's value at exactly the sixth hour of the trading day.

Find the equation of the tangent line to the graph of the function $g(x)$ at the point with abscissa $x = 1$. The function is defined by the following composite expression:

Study the existence of points with a horizontal tangent in the function $f(x) = (x^2 - 4)^2 \times e^{-x^2}$. To do so, obtain the first derivative by applying the chain rule and the product rule, and set it equal to zero to find all possible solutions in the real domain.

Let $f$ and $g$ be two differentiable functions. The following values are known:

• $g(2) = 5$

• $g'(2) = 3$

• $f'(5) = -4$

Use the chain rule to calculate the value of the derivative of the composite function $H(x) = (f \circ g)(x)$ at the point $x = 2$, that is, find $H'(2)$.

Let $f$ and $g$ be two functions for which the following data are known:

• $g(1) = 2$ and $g'(1) = 3$

• $f(2) = -1$ and $f'(2) = 5$

If we define a new function $H(x) = \sqrt{f(g(x)) + 5}$, calculate the value of $H'(1)$ by rigorously applying the chain rule. Comment on what would happen to the derivative if the value inside the square root were negative.

Determine the values of the parameters $a, b \in \mathbb{R}$ so that the following function is continuous and differentiable across its entire domain, paying special attention to the composition in the second branch:

In a study of a monopolistic market, it has been found that the price of a product $p$ (in pesos) and the quantity demanded $q$ (in units) are implicitly related by the following equation:

Knowing that total revenue is given by the function $I = p \times q$, use the chain rule and implicit differentiation to find the derivative of revenue with respect to price, $\frac{dI}{dp}$, when the price is $p = 4$ pesos (consider only the positive value of $q$).