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 $f(x) = (3x - 5) \cdot e^x$, 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 euros) of manufacturing $x$ units of a custom trainer model is given by the function $C(x) = 0,5x^2 + 10x + 200$.

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 $x = 4$.

In a Mathematics Applied to Social Sciences exam, Lucía had to find the derivative function of $f(x) = (x^2 - 3x) \cdot e^x$. On her paper, she wrote the following:

When asked for a justification, she argued: "I differentiated the polynomial inside the parentheses, which gives $2x - 3$, and then I multiplied it by the derivative of $e^x$, which is itself."

1. a) Explain with reasons what Lucía's conceptual error is and which differentiation rule she omitted.
2. b) Correct the exercise by working out step by step the proper derivative using the corresponding rule, and simplify the result by factoring out the common factor.

To calculate the derivative of the rational function $g(x) = \dfrac{4x^5 - 6x^2 + 1}{2x^2}$ on its domain ($x \neq 0$), two classmates propose different strategies:

• Marcos argues that the only correct way is to directly apply the quotient rule for derivatives: $\left(\dfrac{u}{v}\right)' = \dfrac{u'v - uv'}{v^2}$.

• Sara proposes first dividing each term of the numerator by the denominator to transform $g(x)$ into a sum of powers of $x$, and then differentiating term by term.

1. a) Differentiate the function following the method proposed by Sara.
2. b) Justify which of the two methods is more efficient in this specific case, and reason why Marcos's proposal carries a higher risk of algebraic errors.

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) $h(x) = 3^{x^2} \cdot (2x + 5)$

A cosmetics company has estimated that the daily profit (in hundreds of euros) generated by the sale of a new facial serum follows the function $B(t) = 20t \cdot e^{-0,2t}$, 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 $f(x) = x^2 \cdot \ln(x)$, 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).

According to a report on e-commerce in Spain, the number of monthly active customers of a digital platform (in thousands) follows the function $C(t) = 5 + 2t$, where $t$ indicates the month of the year ($t \in [1, 12]$). Meanwhile, the average monthly spending per customer (in euros) decreases slightly throughout the year according to the function $G(t) = 40 \cdot e^{-0,05t}$.

The platform's total monthly revenue (in thousands of euros) is modeled by the product $I(t) = C(t) \cdot G(t)$.

1. a) Apply the product rule to obtain the simplified algebraic expression of the marginal revenue function, $I'(t)$.
2. b) Calculate $I'(6)$ rounded to two decimal places and interpret its socio-economic meaning: is the platform's revenue increasing or decreasing at the end of the first half of the year? Justify your answer.

The percentage of households in a Spanish municipality that have joined a new municipal composting program is modeled by the function $R(t) = \dfrac{80t}{t + 3}$, where $t$ represents the time in months since its launch ($t \ge 0$).

1. a) Before differentiating, estimate the instantaneous growth rate in the first month ($t = 1$) by calculating the Average Rate of Change (A.R.C.) over the interval $[1;\, 1,1]$. Based on this estimation, predict with reasons the sign of $R'(1)$ and whether the rate will exceed $10$ percentage points per month or not.
2. b) Obtain the derivative function $R'(t)$ using the quotient rule, calculate the exact value of $R'(1)$, and check whether your initial estimation and predictions were consistent with the analytical result.

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 modelled the total production cost (in thousands of euros) 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.

The accounting department of an agricultural cooperative estimates that its accumulated profits (in thousands of euros) over time can be modeled by a function with the structure $B(x) = (ax + b) \cdot e^{-0,2x}$, where $x$ measures the quarters elapsed since its re-establishment and the parameters $a, b \in \mathbb{R}$ are unknown constants.

Two data points from the audits are available:

1. At the initial moment ($x = 0$), the cooperative carried a debt of €$10.000$ ($B(0) = -10$).

2. The instantaneous rate of change of the profit at that initial moment ($x = 0$) was positive and exactly equal to $7$ thousand euros per quarter ($B'(0) = 7$).

1. a) Obtain the expression of the derivative function $B'(x)$ in terms of the parameters $a$ and $b$ by rigorously applying the product rule.
2. b) Set up the necessary system of equations and deduce the exact numerical values of $a$ and $b$.
3. c) Find with reasons in which quarter $x$ the marginal profit vanishes ($B'(x) = 0$), and explain the relevance of this point for the growth of the cooperative's earnings.

In a 2nd year of Bachillerato class, a debate arises about how to differentiate the function $f(x) = x^{2x}$ defined for $x > 0$. Two students try to solve it by directly applying rules from the basic table:

• David claims: "I consider the base as a variable and the exponent as a constant; I apply the power rule $(u^n)' = n u^{n-1} u'$, getting $f'_D(x) = 2x \cdot x^{2x - 1} = 2x^{2x}$."

• Elena replies: "No, the base is considered constant and the exponent variable; I apply the exponential rule $(a^u)' = a^u \ln(a) u'$, getting $f'_E(x) = x^{2x} \cdot \ln(x) \cdot 2$."

1. a) Explain why both reasonings are mathematically incorrect and why no row in the elementary table allows this function to be differentiated directly.
2. b) Develop the logarithmic differentiation method to obtain the correct derivative $f'(x)$: take natural logarithms, apply logarithm properties, differentiate both sides using the chain rule and the product rule, and solve for $f'(x)$.
3. c) Compare the correct derivative you obtained with the sum of David's and Elena's results ($f'_D(x) + f'_E(x)$) and formulate a reasoned conclusion regarding this coincidence.