Differentiation Rules

A content creator has observed that the number of views of one of their videos on a social network follows the function $f(t) = t^2 + 20t + 100$, where $t$ is the time in hours since it was published.

Calculate the Average Rate of Change (A.R.O.C.) of the number of views in the interval $[1, 4]$. Interpret the result obtained in the context of the problem.

Calculate the derivative of the following polynomial functions using the basic rules of differentiation:

1. a) $f(x) = 5x^3 - 2x^2 + 7x - 10$

1. b) $g(x) = \frac{1}{4}x^4 + \frac{2}{3}x^3 - 5$

1. c) $h(x) = x^5 - \sqrt{2}x$

The total cost (in pesos) of manufacturing $x$ units of a custom sneaker model is given by the function $C(x) = 10x^2 + 200x + 4\,000$.

1. a) Calculate the marginal cost function $C'(x)$.

1. b) Find the marginal cost when 20 units are produced and interpret what this value means for the company.

Given the function $f(x) = (3x - 5) \times e^x$, find its derivative by applying the product rule. Simplify the result by factorising where possible.

Calculate the derivative of the following rational function using the quotient rule:

State for which values of $x$ the derivative of this function does not exist, based on its domain.

Find the equation of the tangent line to the graph of the function $f(x) = x^2 - 4x + 1$ at the point with abscissa $x = 3$.

The figure shows the graph of $f$ and the point $P$ on the curve with that abscissa.

Given the function $f(x) = \dfrac{x^2 + k}{x - 1}$, calculate the value of the constant $k$ knowing that the gradient of the tangent line to the graph of $f(x)$ at the point with abscissa $x = 2$ is equal to $-3$. Once $k$ has been found, write the complete equation of this tangent line.

Consider the curve $y = x^3 - 6x^2 + 4$ and the line $r: y = -9x + 2$.

1. a) Find the points of the curve at which the tangent line is parallel to the line $r$.

1. b) Write the equations of the tangent lines to the curve at those points.

Find the equations of the tangent lines to the curve $f(x) = x^2 - 4x + 3$ that pass through the external point $P(2, -5)$.

Hint: Remember that point $P$ does not necessarily lie on the graph of the function. You should write the gradient of the tangent line at a generic point $(a, f(a))$ and equate it to the gradient of the line connecting that point to $P$.

Given the function $f(x) = \sqrt{x^2 + 9}$, calculate its derivative using the chain rule. Once obtained, find the value of the gradient of the tangent line to the curve at the point $x = 4$.

Apply the chain rule to find the derivative of the following composite functions:

1. a) $f(x) = (2x^2 + 1)^4$

1. b) $g(x) = \ln(5x - 2)$

1. c) $h(x) = e^{3x^2}$

Calculate the derivative of the following functions using the rules of differentiation, simplifying the final result whenever possible:

1. a) $f(x) = \sqrt[3]{(5x^2 - 1)^2}$

1. b) $g(x) = \ln\left(\dfrac{x^2 + 3}{x - 2}\right)$ (use the properties of logarithms before differentiating).

1. c) $h(x) = 3^{x^2} \times (2x + 5)$

A cosmetics company has estimated that the daily profit (in hundreds of pesos) generated by the sale of a new facial serum follows the function $B(t) = 20t \times e^{-0.2t}$, where $t$ is the time in months since its launch. The figure shows the graph of $B(t)$ and marks the point corresponding to $t=5$.

1. a) Obtain the expression of the function $B'(t)$, which represents the rate of change of the profit.

1. b) Calculate the value of the derivative 5 months after its launch and interpret the result obtained in economic terms.

Consider the function $f(x) = \dfrac{e^{2x}}{x^2 + 1}$.

1. a) Calculate its derivative $f'(x)$, simplifying the expression as much as possible.

1. b) Determine whether there is any point where the curve has a horizontal tangent (zero gradient). If so, state the coordinates of that point.

A biotechnology company has modeled the total production cost (in thousands of pesos) of a new drug using the function:

where $x$ is the number of batches produced.

1. Calculate the marginal cost function $C'(x)$.

2. Determine the rate at which the cost changes when exactly 4 batches are being produced.

Determine the values of the parameters $a$ and $b$ so that the following function is continuous and differentiable across the set of real numbers $\mathbb{R}$:

$f(x) = \begin{cases} ax^2 + 3x & \text{if } x \le 2 \\ \dfrac{b}{x} + 1 & \text{if } x > 2 \end{cases}$

Given the following piecewise function, determine the values of the parameters $a$ and $b$ so that the function is differentiable across its entire domain (especially at the boundary point $x = 1$):

Apply the technique of logarithmic differentiation to find the expression for the derivative of:

$f(x) = (x^2 + 1)^{\ln(x)}$

Given the function $f(x) = x^2 \times \ln(x)$, calculate:

1. a) The first derivative $f'(x)$.

1. b) The second derivative $f''(x)$.

1. c) The equation of the normal line to the graph at the point with abscissa $x = e$. (Remember that the normal line is perpendicular to the tangent at the point of contact).

Given the rational function $f(x) = \frac{1}{2x - 3}$, obtain a general expression for its nth derivative, $f^{(n)}(x)$.

To do this, calculate the 1st, 2nd, 3rd, and 4th order derivatives, identify the pattern in numbers and signs, and write it in terms of $n$ and the factorial $n!$.

Determine the values of $a, b \in \mathbb{R}$ so that the function $f(x) = \dfrac{ax + b}{x^2 - 4}$ has a relative extremum at the point $P(0, -1)$. Once you have found the parameters, use the second derivative to confirm whether it is a maximum or a minimum.

The number of active users on a new social network (in millions) follows a logistic growth model given by the function:

where $t$ is the time in months since its launch.

1. Find the function $N'(t)$ representing the growth rate of the social network.

2. In which month is the user growth rate at its maximum?

Hint: You must find the maximum of the derivative function $N'(t)$, which involves calculating the second derivative $N''(t)$ and setting it equal to zero.