The Derivative Function

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

Using only the definition of the derivative as a limit (difference quotient), calculate the derivative function of $f(x) = \sqrt{2x^2 + 1}$. Then, use the result to find the equation of the tangent line to the curve at the point $P$ with abscissa $x = 2$.

The figure shows the graph of $f$ and the position of the point $P$.

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

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 $f(x) = x \times \ln(x)$, calculate its derivative $f'(x)$ applying the product rule and determine the gradient 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 how the incline of the ramp varies at any point along 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 abscissa $x = 3$.

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

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 satisfied:

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 $f''(0) = 6$.

A surveillance drone follows a path whose height in meters 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 times is the drone instantaneously at rest (velocity equal to zero)?

A surveillance drone follows a flight path along a straight line. Its position (in decameters) relative to the control center is given by the function $s(t) = t^2 \times e^{-t/4}$, where $t$ is the time in minutes since takeoff ($t \ge 0$).

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

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

Study the differentiability of the function $f(x) = |x^2 - 4x + 3|$ at the points $x = 1$ and $x = 3$.

1. a) Study whether $f$ is continuous at $x = 1$ and at $x = 3$.

1. b) Calculate, if they exist, the one-sided derivatives of $f$ at $x = 1$ and at $x = 3$.

1. c) State whether $f$ is differentiable at those points.

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 $f: \mathbb{R} \to \mathbb{R}$ defined 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 $x = 2$.

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

The following graph corresponds to the derivative function, $f'(x)$, of a function $f(x)$ that is continuous on its entire domain $\mathbb{R}$. The graph of $f'(x)$ is a parabola that crosses the $X$-axis at $x=-2$ and $x=2$, and whose vertex is the point $(0,2)$.

Using the graph of $f'(x)$, answer the following questions about the original function $f(x)$, justifying your reasoning:

1. a) State the intervals of increase and decrease of $f(x)$.

1. b) Determine the $x$-coordinates of the relative extrema of $f(x)$ and classify them as maxima or minima.

1. c) At which value of $x$ does $f(x)$ have an inflection point? Justify your answer based on the relationship between $f''(x)$ and the slope of $f'(x)$.

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

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

1. a) Decompose the function into partial fractions.
2. b) Using the above 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.

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

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

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

State with reasoning 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 gradient of the tangent line is zero?

Given the equation of the curve implicitly defined by $x^3 + y^3 - 3xy = 0$ (known as the Folium of Descartes), calculate the value of the gradient 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), answer the following:

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 $( \frac{3}{2}, \frac{3}{2} )$.