The Derivative as a Function

A skater moves along a track following a trajectory whose position (in feet) is given by the function $s(t) = t^2 + 3t$, where $t$ is the time in seconds. Calculate, using the definition of the derivative as a limit, the instantaneous velocity of the skater at the instant $t = 2$ seconds.

Making exclusive use of the definition of the derivative as a limit (difference quotient), calculate the derivative function of $f(x) = \sqrt{2x^2 + 1}$. Subsequently, use the result obtained to find the equation of the tangent line to the curve at the point with x-coordinate $x = 2$.

Calculate the derivative function of the following elementary functions using the rules of differentiation:

1. a) $f(x) = 5x^4 - 3x^2 + 7x - 1$

1. b) $g(x) = \frac{1}{x^3} + \sqrt{x}$

1. c) $h(x) = 3e^x - 2\cos(x)$

Given the function $f(x) = x \cdot \ln(x)$, calculate its derivative $f'(x)$ applying the product rule and determine the value of the slope of the curve at the point where $x = e$.

An access ramp for a skatepark is being designed. The side profile of the first section follows the function $f(x) = \frac{x^2+4}{x+1}$. Calculate the derivative function $f'(x)$ to find out how the incline of the ramp varies at any point in that section.

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

1. a) $f(x) = (3x^2 - 5)^4$

1. b) $g(x) = \sin(5x + 2)$

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

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

$f(x) = \ln\left(\frac{1 + \sin(x)}{1 - \sin(x)}\right)$

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

Find the equation of the tangent line to the curve $f(x) = \frac{x^2 - 1}{x + 2}$ at the point where the function intersects the y-axis ($OY$).

Calculate the second derivative of the function $f(x) = x^3 - 6x^2 + 9x - 2$. For what value of $x$ is this second derivative zero? (This point is important for studying concavity, which you will see later.)

Given the function $f(x) = \ln(2x - 3)$, find the general expression for its n-th derivative $f^{(n)}(x)$ and use it to calculate the value of the fourth derivative at the point $x = 2$.

Consider the function $f(x) = \frac{1}{x^2 - 3x + 2}$.

1. a) Decompose the function into partial fractions.
2. b) Using the previous decomposition, find a general formula for the $n$-th derivative, $f^{(n)}(x)$, for any $n \in \mathbb{N}$.
3. c) Prove the validity of your formula for every $n \in \mathbb{N}$ by mathematical induction, explicitly checking the first three orders of differentiation.

A surveillance drone follows a trajectory whose height in feet is given by the function $h(t) = \frac{1}{3}t^3 - 4t^2 + 11t + 5$, where $t$ is the time in seconds ($0 \leq t \leq 10$).

• a) Calculate the velocity function $v(t)$ and the acceleration function $a(t)$.

• b) At what instants of time is the drone momentarily stationary (velocity equal to zero)?

A surveillance drone follows a flight path along a straight axis. Its position (in feet) with respect to the control center is given by the function $s(t) = t^2 \cdot e^{-t/4}$, where $t$ is the time in minutes since take-off ($t \ge 0$).

1. a) Determine the velocity function $v(t)$ and the acceleration function $a(t)$ of the drone.
2. b) At what instant does the drone change its direction of motion?
3. c) Calculate the exact moment when the acceleration is zero. What physical interpretation does this point have with respect to the drone's velocity?

The following graph corresponds to the derivative function, $f'(x)$, of a function $f(x)$ which is continuous throughout its domain $\mathbb{R}$.

Based on the graph of $f'(x)$, answer the following questions about the original function $f(x)$, giving reasons:

1. a) State the intervals of increase and decrease of $f(x)$.
2. b) Determine the x-coordinates of the relative extrema of $f(x)$ and classify them as maxima or minima.
3. c) At what value of $x$ does $f(x)$ have a point of inflection? Justify your answer based on the relationship between $f''(x)$ and the slope of $f'(x)$.

Consider the function $f(x) = x^4 + ax^3 + bx^2 + cx + d$. Determine the coefficients $a, b, c$ and $d$ knowing that the following conditions are simultaneously met:

1. The graph of $f(x)$ passes through the origin.

2. The function has a point of inflection at $x = 1$.

3. The tangent line to the curve at the point with x-coordinate $x = 0$ is parallel to the line $y = -2x + 5$.

4. The value of the second derivative at $x = 0$ is $f''(0) = 6$.

Determine the differentiability of the following function at the point $x = 2$, first checking if it is continuous at that point:

Study the differentiability of the function $f(x) = |x^2 - 4x + 3|$ at the points $x = 1$ and $x = 3$. State whether one-sided derivatives exist at these points and whether the function is continuous.

Determine the values of the parameters $a$ and $b$ so that the following function is differentiable throughout its domain, paying special attention to the joining point $x = 0$:

$f(x) = \begin{cases} e^{ax} & \text{if } x \leq 0 \\ x^2 + bx + 1 & \text{if } x > 0 \end{cases}$

Consider the piecewise function $f: \mathbb{R} \to \mathbb{R}$ given by the following expression:

Determine the values of the parameters $a, b \in \mathbb{R}$ so that the function is differentiable throughout its domain. Once found, discuss whether the derivative function $f'(x)$ is, in turn, continuous at $x = 2$.

Given the equation of the implicitly defined curve $x^3 + y^3 - 3xy = 0$ (known as the Folium of Descartes), calculate the value of the slope of the tangent line to the curve at the point $( \frac{3}{2}, \frac{3}{2} )$.

Given the curve defined by the implicit equation $x^3 + y^3 - 3xy = 0$ (known as the Folium of Descartes):

1. a) Obtain the expression for the derivative $y'$ in terms of $x$ and $y$ using implicit differentiation.
2. b) Find the points on the curve where the tangent line is horizontal.
3. c) Determine the equation of the normal line to the curve at the point $( \frac{3}{2}, \frac{3}{2} )$.

Use the technique of logarithmic differentiation to find the derivative of the following power-exponential function:

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

Use the technique of logarithmic differentiation to calculate the derivative function of:

State, with justification, the domain of definition of the function $f(x)$ and that of its derivative function $f'(x)$. Is there any point in the interval $(0, \pi/2)$ where the slope of the tangent line is zero?