The derivative function

A skater moves along a track following a trajectory whose position (in metres) 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.

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 $x = e$.

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

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 $x = 2$.

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

A surveillance drone follows a path whose height in metres 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)?

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 $f''(0) = 6$.

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).

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 $x = 2$.

We want to determine the real parameters $a$ and $b$ of a rational function of the form $f(x) = \frac{a x + b}{x - 2}$ (with $x \ne 2$) so that it simultaneously satisfies two conditions:

1. Its $n$-th derivative satisfies, for any natural number $n \ge 1$, the general expression:

1. The tangent line to the graph of $f$ at its $x$-intercept forms an angle of $135^\circ$ with the positive direction of the $OX$ axis.

1. a) Deduce by pattern recognition or recurrence the formula for the $n$-th order derivative of the elementary function $g(x) = \frac{1}{x - 2}$.
2. b) Decompose $f(x)$ into the form $f(x) = A + \frac{B}{x - 2}$ and use condition 1 to fix the value of $B$, obtaining a first relation between $a$ and $b$.
3. c) Analytically translate geometric condition 2 about the tangent line to find all pairs of values $(a, b)$ that satisfy the statement, checking the consistency of the result in each case.

Aoife is calculating the derivative of the function $f(x) = x^{2x+1}$ for $x > 0$. On her exam, she writes the following solution:

1. a) Explain in detail the conceptual error Aoife makes and why the usual differentiation rules for power functions ($x^k$) or exponential functions ($a^x$) cannot be applied to this function.
2. b) Correct the exercise by applying the logarithmic differentiation method step by step to obtain the correct expression for $f'(x)$, leaving it simplified and expressed exclusively in terms of $x$.

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