Exercises: 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) $h(x) = x^5 - \sqrt{2}x$

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.

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

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 $f(x) = \sqrt{x^2 + 9}$, 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 $x = 4$.

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.

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 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.

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

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

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.

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).

Calculate the derivative function of the following expression using logarithmic differentiation. Take into account that the base and the exponent are functions of $x$:

Show all intermediate steps, applying the laws of logarithms correctly before differentiating.

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 $x = 1$):

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 $P$.

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.

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 $n!$.

Calculate the derivative of the following composite function, simplifying the final result as much as possible:

Note: Using trigonometric identities after differentiating can help you simplify the expression surprisingly well.

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.