Exercises: Derivative function. Higher-order derivatives

Given the function $f(x) = 3x^2 - 5x + 2$, carry out the following tasks:

1. a) Calculate the derivative function $f'(x)$ by applying the differentiation rules.

1. b) Find the value of the slope of the tangent line to the graph of $f(x)$ at the point with abscissa $x = 2$.

1. c) Determine the equation of the tangent line at that point.

A phone case manufacturing company has determined that the total cost (in euros) of producing $x$ units is given by the function:

Calculate the marginal cost function, $C'(x)$, and determine the approximate cost of producing the 21st unit (that is, calculate $C'(20)$).

Calculate the first three derivatives of the polynomial function $f(x) = x^4 - 2x^3 + 5x - 7$. Once obtained, check whether $f'''(1) = 18$ holds.

The position of a delivery drone relative to its takeoff base follows the function $s(t) = t^2 + 4t$, where $s$ is the distance in metres and $t$ is the time in seconds.

1. Obtain the function for instantaneous velocity, $v(t)$, knowing that it is the first derivative of position.

2. Obtain the acceleration function, $a(t)$, knowing that it is the second derivative of position.

3. What speed does the drone have 5 seconds into the flight?

Find the points on the graph of the function $f(x) = x^3 - 12x + 1$ at which the tangent line is horizontal.

Hint: A line is horizontal when its slope is equal to $0$.

Given the function $f(x) = \frac{x^2}{2} + 4\sqrt{x}$, calculate its derivative function $f'(x)$. Once calculated, find the value of $f'(4)$.

It is known that the derivative of a function at a point is equal to the gradient of the tangent line. If the tangent line to the function $g(x)$ at the point $x = 3$ is $y = 4x - 5$, explain what the values of $g(3)$ and $g'(3)$ are.

Verify that the fourth derivative of the function $f(x) = \frac{1}{6}x^3 + x^2$ is equal to zero for any value of $x$. What is the minimum degree a polynomial must have for its fourth derivative to be nonzero?

Marcos manages an artisan footwear workshop in Elche. The function describing the total daily cost (in euros) of manufacturing $x$ pairs of shoes is $C(x) = 200 + 15x + 0,05x^2$. In order to budget for a staff increase, Marcos wants to know approximately how much producing the 51st pair will cost. To do so, he performs the following calculation:

and concludes: "Manufacturing the 51st pair will cost us exactly 20,10 euros, since by definition the marginal cost $C'(51)$ is equal to the difference $C(51) - C(50)$".

1. a) Explain the two conceptual errors made by Marcos: one regarding the point at which the marginal function should be evaluated to estimate the effect of the 51st unit, and another regarding the word "exactly".
2. b) Correct the calculation: evaluate the marginal cost function at the appropriate value of $x$ and interpret the result obtained.
3. c) Calculate the exact actual cost of producing that additional unit using the difference $C(51) - C(50)$ and compare it with the marginal estimate from the previous part. Is it a good approximation?

The temperature $T(t)$ (in degrees Celsius) recorded by an AEMET weather station in Córdoba during a hot summer day, from 8:00 ($t=0$) to 20:00 ($t=12$), is modeled by the function:

1. a) Before carrying out any analytical calculation, observe the function (a concave parabola with a vertex at $t=7,5$, that is, at 15:30). Reasonably estimate whether the heating rate at 10:00 ($t=2$) will be greater or less than at 14:00 ($t=6$), and whether the derivative $T'(2)$ should be positive or negative.
2. b) Calculate the first derivative function $T'(t)$ and evaluate it at $t=2$ and at $t=6$. Check whether the obtained values confirm your previous estimate and interpret the meaning of their signs and their units of measurement.
3. c) Find the second derivative function $T''(t)$. Interpret its constant value and explain what information it provides about how the warming rate of the air evolves throughout the daytime hours.

Given the polynomial function $f(x) = \frac{1}{12}x^4 - \frac{2}{3}x^3 + \frac{5}{2}x^2 - 10$, perform the following operations:

1. a) Calculate the first derivative $f'(x)$ and the second derivative $f''(x)$.
2. b) Determine the values of $x$ for which the second derivative is zero ($f''(x) = 0$).
3. c) Calculate the higher-order derivative $f^{(5)}(x)$ and justify the result obtained without needing to carry out all intermediate steps.

A courier company has checked that the total delivery cost (in hundreds of euros) follows the function $C(x) = \sqrt{x^2 + 9}$, where $x$ is the number of thousands of packages sent.

1. a) Find the marginal cost function, i.e. the derivative of the cost $C'(x)$.
2. b) Calculate the rate at which the cost is increasing when $4000$ packages are being sent ($x=4$).
3. c) Calculate the second derivative $C''(x)$, simplifying the expression as much as possible.

It is known that the function $f(x) = ax^3 + bx^2 + 9x$ has a second derivative that equals zero at $x = 2$. Furthermore, it is known that the value of the first derivative at the point $x = 1$ is equal to $6$. Calculate the values of the real parameters $a$ and $b$.

Given the function $g(x) = e^{2x} \cdot (x^2 - 1)$, prove that the following relationship holds between the function and its successive derivatives for any value of $x$:

The number of followers of an influencer on a social network grows according to the function $S(t) = \frac{50t}{t+2}$, where $t$ is the time in months since opening the account and $S(t)$ is measured in thousands of followers.

1. a) Obtain the derivative function $S'(t)$, which represents the growth rate of the community.
2. b) At what rate is the account gaining followers exactly at the end of the fourth month?
3. c) Study the sign of the second derivative $S''(t)$ for $t > 0$ and explain what this result means in terms of the growth of the account (is it growing faster and faster or slower and slower?).

Find the values of $k$ and $p$ so that the function $f(x)$ is differentiable on its entire domain (especially at the jump point $x = 0$):

Once found, calculate the value of $f''(1)$.

Given the function $h(x) = \ln(x^2 + 1)$, whose graph appears in the figure, calculate:

1. a) The slope of the tangent line to the curve at the point with abscissa $x = 1$.

1. b) The equation of the second derivative function $h''(x)$.

1. c) The points of the graph where the second derivative is equal to zero.

Consider the function $f(x) = \frac{x+1}{x-1}$.

1. a) Calculate $f'(x)$, $f''(x)$ and $f'''(x)$.
2. b) By observing the previous results, try to deduce a general expression for the $n$th derivative $f^{(n)}(x)$.

(Hint: Look at how the exponent in the denominator and the sign of the fraction evolve).

To calculate the successive derivatives of the rational function $f(x) = \dfrac{2x^3 - 4x^2 + 6x}{2x}$, two 2nd-year Bachillerato students propose different approaches:

• Irene proposes applying the quotient rule directly to obtain $f'(x)$, and applying the quotient rule again to the result to find $f''(x)$.

• Lucas argues that it is much faster and safer to simplify the expression by dividing each term of the numerator by the denominator before starting to differentiate.

1. a) Compare both methods and justify which one is more convenient. Specify the domain of definition of $f(x)$ and clarify whether Lucas's simplification is valid for all points in that domain.
2. b) Using the most efficient method, calculate the simplified expressions of the derivative function $f'(x)$, the second derivative $f''(x)$, and the third derivative $f'''(x)$.
3. c) From the results obtained, deduce in a reasoned manner the value of the tenth-order derivative, $f^{(10)}(x)$, for any point in the domain, justifying your conclusion based on the degree of the function without performing intermediate calculations.

A telecommunications company operates in Spain offering a monthly fiber and mobile bundle. The analytics department estimates that if it sets a monthly fee of $p$ euros ($p \in [10, 60]$), the number of subscribers in thousands will be given by the demand function $q(p) = 60 - 0,8p$. The monthly fixed infrastructure costs are $150$ thousand euros and the variable customer service and support cost is $10$ euros per month per subscribed customer (that is, $10$ thousand euros per thousand customers).

1. a) Write the analytical expression of the revenue function $I(p)$ and the total cost function $C(p)$, both in thousands of euros and solely as a function of the price $p$. From them, obtain the monthly profit function $B(p) = I(p) - C(p)$.
2. b) Calculate the marginal profit function with respect to price, $B'(p)$, and the second derivative function $B''(p)$.
3. c) Interpret the constant sign of $B''(p)$ in economic terms: does profit increase at an ever-increasing or ever-decreasing rate as the fee becomes more expensive?
4. d) Determine the price $p$ for which marginal profit is zero ($B'(p) = 0$). Explain what happens to the company's overall profit if it decides to set a fee lower or higher than that price.

Determine the general expression for the $n$-th derivative of the rational function $f(x) = \dfrac{1}{3x - 2}$, defined for $x \neq \dfrac{2}{3}$. To do so, calculate the first four successive derivatives, identify the recurring pattern, and use factorial notation $n!$ and powers to write $f^{(n)}(x)$.

A mobile app development company has found that the number of active users of its new game, measured in thousands, follows the function $U(t) = t^2 \cdot e^{-0,5t}$, where $t$ is the time in months since launch. The graph of the function is as follows:

The "acceleration" in user acquisition is given by the second derivative of the function, $U''(t)$.

1. a) Calculate the second derivative function $U''(t)$.

1. b) Determine at which moments (months) the acceleration is zero (inflection points of the growth).

Given the function $f(x) = a \cdot \ln(x) + b \cdot x^2 + x$, calculate the values of the real parameters $a$ and $b$ knowing that the first derivative function equals zero at $x = 1$, that is, $f'(1) = 0$, and that the second derivative at that same point equals $f''(1) = 5$. Once $a$ and $b$ have been found, write the expression for the third derivative $f'''(x)$.

In the analysis of computer algorithms, the complexity of a cost function defined by $C(x) = x^x$ for $x > 0$ is studied. Use the technique of logarithmic differentiation to find the derivative $C'(x)$. Subsequently, calculate the second derivative $C''(x)$ and check whether there is any value of $x$ for which the derivative of the gradient is zero.

Consider the composite exponential function $g(x) = e^{x^2 - 4x}$. Find the successive derivatives $g'(x)$ and $g''(x)$. Find the points on the graph where the original function and its second derivative take the same value, that is, solve the equation $g(x) = g''(x)$.

A sociologist studies the spread of a rumour on a social network using the function $P(t) = \dfrac{100}{1 + 9e^{-t}}$, where $P$ is the percentage of the population that knows the rumour and $t$ is the time in days.

Show that the speed of propagation (the first derivative $P'(t)$) is at its maximum when the second derivative is zero ($P''(t) = 0$). Calculate this exact time $t$ using natural logarithms.

Calculate the values of the parameters $k$ and $p$ so that the function $f(x) = \sin(kx) + e^{px}$ simultaneously satisfies the following conditions:

1. Its first derivative at the origin is zero: $f'(0) = 0$.

2. Its second derivative at the origin is equal to 2: $f''(0) = 2$.

Justify whether the values obtained are unique or whether multiple combinations exist.

Given the function $f(x) = \sqrt{x^2 + k}$, where $k$ is a positive constant, show that the following relationship holds:

To do this, first obtain the derivative $f'(x)$, simplify it, and then calculate the second derivative $f''(x)$. Substitute both into the left-hand side expression to verify the equality.

A financial consulting team needs to determine the total cost function $C(x)$ (in euros) of an electric vehicle component factory in Almussafes, where $x$ represents the number of units produced daily. It is known that the behavior of costs corresponds to a third-degree polynomial of the form $C(x) = ax^3 + bx^2 + cx + d$, and that the following four conditions are simultaneously met:

1. Keeping the plant open without producing any units ($x=0$) generates daily fixed costs of $1200$ euros.

2. Upon reaching a production of $10$ units per day, the marginal cost is $40$ euros per unit.

3. The second derivative of cost vanishes exactly for a production volume of $x = 20$ units ($C''(20) = 0$), the point from which the growth of costs stops decelerating and begins to accelerate.

4. For that same level of $20$ units per day, the marginal cost reaches its absolute minimum value, which is $10$ euros per unit.

1. a) Translate each of the four conditions into an equation involving $C(x)$, $C'(x)$, or $C''(x)$.
2. b) Set up and solve the resulting system to find the numerical values of the parameters $a$, $b$, $c$, and $d$.
3. c) Once the function $C'(x)$ is obtained, prove algebraically that the marginal cost $C'(x)$ is strictly positive for any production level $x \ge 0$. What real economic implication does this fact have for the factory?

To study the impact of an awareness campaign by the Dirección General de Tráfico (DGT) on social media, the proportion of young drivers reached is modeled by the function $f(t) = \dfrac{t}{t+2}$, where $t \ge 0$ is the time elapsed in weeks.

To predict the long-term behavior of the successive accelerations of the diffusion, two analysts attempt to deduce the general formula for the $n$-th derivative, $f^{(n)}(t)$:

• Lucía proposes the formula: $f^{(n)}(t) = \dfrac{(-1)^{n-1} \cdot 2 \cdot n!}{(t+2)^{n+1}}$, for all $n \ge 1$.

• Sergio argues that the correct formula is: $f^{(n)}(t) = \dfrac{(-1)^n \cdot n!}{(t+2)^{n+1}}$, for all $n \ge 1$.

1. a) Verify that $f(t)$ can be rewritten algebraically as $f(t) = 1 - \dfrac{2}{t+2}$. Justify why using this expression significantly simplifies the calculation of successive derivatives compared to repeatedly differentiating as a quotient.
2. b) Sequentially calculate the first four derivative functions: $f'(t)$, $f''(t)$, $f'''(t)$, and $f^{(4)}(t)$, expressing each as a single fraction without negative exponents.
3. c) Analyze rigorously who is right between Lucía and Sergio. To do so, examine in detail the three patterns involved in the $n$-th derivative: the alternating sign rule, the numerical coefficient (and the appearance of the factorial $n!$), and the exponent of the denominator. Point out precisely the errors made by the incorrect person.
4. d) From the second derivative $f''(t)$, determine its sign for any instant $t > 0$ and interpret the result in the context of the campaign: is the rate at which new informed drivers join accelerating or slowing down over the weeks?