The Derivative Function. Successive Derivatives

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.

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.

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

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 $g(x) = g''(x)$.

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 phone case manufacturing company has determined that the total cost (in pesos) 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 $C'(20)$).

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 meters 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?

A courier company has checked that the total delivery cost (in hundreds of pesos) 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.

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.

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?

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.

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?).

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.

A mobile app development company has found that the number of active users of its new game, measured in thousands, follows the function $U(t) = t^2 \times e^{-0.5t}$, where $t$ is the time in months since launch.

The "acceleration" in user acquisition is given by the second derivative of the function, $U''(t)$. Calculate the second derivative function and determine in which month the acceleration is zero (the inflection point of the growth).

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.

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 $b$.

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 $f(x) = a \times \ln(x) + b \times x^2 + x$, 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 $f'''(x)$.

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.

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).

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 $f^{(n)}(x)$.

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.