Exercises: Derivative function

Apply the definition of the derivative using the limit calculation $f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$ to find the derivative function of $f(x) = 3x^2 - 5x + 2$.

Calculate the derivative function of the following polynomial and exponential function, applying the basic rules of differentiation:

$f(x) = \frac{3}{4}x^4 - 2x^3 + \frac{5}{x^2} - 3e^x$

Obtain the derivative function of $f(x)$ applying the product rule. Simplify the result as much as possible.

$f(x) = (2x^3 - x + 4) \times \ln(x)$

Find the derivative function applying the quotient rule. Express the final result with the numerator simplified.

$f(x) = \frac{4x - 3}{x^2 + 5}$

Use the chain rule to determine the derivative function of the following composite trigonometric function:

$f(x) = \cos(5x^3 - 2x^2 + 1)$

Calculate the exact value of the derivative of the function $f(x) = \sqrt{3x^2 + 4}$ evaluated at the point with x-coordinate $x = 2$.

Determine the second derivative, $f''(x)$, of the function $f(x) = x^4 \times \ln(x)$.

Observe the graph of the function $f(x)$ representing the trajectory of a projectile as a function of time.

From the visual representation, determine the sign of the derivative function $f'(x)$ at the x-coordinates $x = 1$, $x = 2$, and $x = 3$. Justify your answer based on the geometric concept of the derivative.

The equation of motion of an autonomous test vehicle is given by the function $s(t) = 2t^3 - 15t^2 + 24t + 10$, where $s$ is the position in meters and $t$ is the time in seconds ($t \ge 0$).

Find the velocity function $v(t)$ and the acceleration function $a(t)$. Determine at what times the vehicle comes to a momentary standstill.

The total production cost, in pesos, of $x$ units of a high-precision electronic component is modeled by the function $C(x) = 0.40x^3 - 12x^2 + 800x + 10\,000$.

Calculate the marginal cost function $C'(x)$ and evaluate its value when exactly $x = 15$ units are produced. Interpret the result obtained in the context of the problem.

In a chemical kinetics experiment, the concentration of a reactant as a function of time $t$ (in minutes) fits the expression $C(t) = \frac{8t}{t^2 + 4}$ in moles per liter.

Find the derivative function $C'(t)$ that represents the reaction rate. At what time $t > 0$ is the reaction rate zero?

Determine the values of the real parameters $a$ and $b$ such that the following piecewise-defined function is differentiable at $x = 2$:

$f(x) = \begin{cases} ax^2 + 3x & \text{if } x \le 2 \\ bx^3 - 4 & \text{if } x > 2 \end{cases}$

Apply the properties of logarithms to simplify the expression for $f(x)$ before differentiating, and then calculate its simplified derivative function:

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

Find the x-coordinates of the points on the curve $f(x) = e^{3x}(x^2 - 4x + 2)$ at which the tangent line is horizontal.

The cumulative number of views of a viral video $t$ hours after its publication follows an exponential growth law given by $R(t) = 1500 \times 3^{0.4t}$.

Obtain the function describing the instantaneous growth rate of views, $R'(t)$, and calculate this rate after $5$ hours.

The following graph represents the derivative function $f'(x)$ of a continuous function $f(x)$ across all of $\mathbb{R}$.

Observing exclusively the behavior of $f'(x)$, determine the intervals where the original function $f(x)$ is increasing and where it is decreasing. At which x-coordinates will $f(x)$ have relative extrema?

Find the general expression for the nth derivative, $f^n(x)$, of the function $f(x) = \ln(3x - 2)$. Hint: calculate the first four successive derivatives and deduce the pattern.

Consider the function defined using a modulus $f(x) = |x^2 - 4|$.

Express $f(x)$ as a piecewise function and calculate its derivative function $f'(x)$. Prove analytically why the function is not differentiable at the corner points indicated on the graph.

Show that the function $y = e^{-2x} \times \sin(3x)$ is a particular solution to the second-order linear differential equation:

$y'' + 4y' + 13y = 0$

Given the rational function $f(x) = \frac{ax^2 + b}{cx + 1}$, determine the values of the real parameters $a, b, c$ knowing that its graph passes through the point $(0, 3)$, has a vertical asymptote at $x = -2$, and has a horizontal tangent line at the point with x-coordinate $x = 1$.

In the design of a suspension bridge, the profile of the main cable takes the form of a catenary whose equation in a coordinate system is $f(x) = \frac{e^{x/10} + e^{-x/10}}{2}$ (with $x$ and $y$ in meters).

Calculate the angle of inclination $\alpha$ that the tangent line to the cable makes with the horizontal at the anchorage point located at $x = 10$ meters.

Consider the function $f(x) = \begin{cases} e^{ax} - 1 & \text{if } x \le 0 \\ b \sin(x) + c x^2 & \text{if } x > 0 \end{cases}$.

Determine the mathematical relationship that must exist between the parameters $a, b, c$ for the second derivative to exist at the origin, that is, for $f''(0)$ to be a well-defined real value.

The following panel shows two curves, A (blue) and B (green). It is known that one of them corresponds to a polynomial function $f(x)$ and the other to its derivative function $f'(x)$.

Give reasons to identify which curve represents $f(x)$ and which represents $f'(x)$ based on the geometric relationship between the intercepts, the relative extrema, and the intervals of increase/decrease. Next, deduce the sign of the second derivative $f''(x)$ at the x-coordinate $x = 0$.