Differentiation Rules and Derivatives of Common Functions

Calculate the derivative function of the following functions by applying the differentiation rules for elementary functions (powers and roots):

1. a) $f(x) = x^7 - 5x^3 + \pi$

1. b) $g(x) = \sqrt{x} + \frac{1}{x^2}$

Find the derivative function of the following transcendental functions using standard derivative tables:

1. a) $f(x) = 4 \times \ln(x) + \log_{2}(x)$

1. b) $g(x) = 3^x - 5 \times \sin(x)$

Given the function $f(x) = \frac{x^4}{4} - \sqrt[3]{x}$, calculate its derivative function $f'(x)$ and simplify the result, writing the root back in radical form.

A fireworks rocket takes off vertically from a launch pad. The height $h$ in meters with respect to time $t$ in seconds is given by the polynomial function $h(t) = t^3 - 4t^2 + 5t + 2$. Calculate the derivative function $h'(t)$, which represents the instantaneous velocity of the rocket at any instant $t$.

Given the quadratic function $f(x) = x^2 - 6x + 5$, find:

1. a) The derivative function $f'(x)$.

1. b) The value of the slope of the tangent line to the graph of the function at the point $P$ with abscissa $x = 4$ (marked in the figure).

A spring oscillates following simple harmonic motion described by the function $y(x) = 5 \times \cos(x) + 2$, where $y$ is the position in centimeters and $x$ is time. Obtain the expression for the derivative function $y'(x)$, which represents the oscillation velocity of the spring.

A social media account has analysed its growth and determined that the number of followers $S$ evolves according to the exponential function $S(t) = 150 \times e^{0.4t}$, where $t$ is the elapsed time in months since its creation. Determine the derivative function $S'(t)$ to find the rate of follower growth.

Calculate the derivative of the following simple composite functions by applying the chain rule directly:

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

1. b) $g(x) = e^{x^2 - 1}$

Calculate the derivative function of the following composite function, applying the chain rule and simplifying the result as much as possible:

$f(x) = \cos^3(e^{x^2+1})$

Calculate the equation of the tangent line to the curve $f(x) = \arctan\left(\frac{x}{2}\right)$ at the point with abscissa $x = 2$.

The figure shows the graph of $f$, its horizontal asymptotes and the point $P$ of the curve whose abscissa is $x = 2$.

A surveillance drone follows a trajectory whose height in meters is given by the function $h(t) = 15t \times e^{-0.2t}$, where $t$ is the time in seconds since takeoff.

1. a) Calculate the function representing the vertical velocity of the drone, $v(t) = h'(t)$.
2. b) What is the vertical velocity of the drone at the instant $t = 5$ seconds?

Given the rational function involving natural logarithms:

$g(x) = \frac{\ln(x^2 + 4)}{x}$

Find the expression of its derivative $g'(x)$ and determine the values of $x$ for which the gradient of the tangent line to the graph of $g(x)$ is equal to zero.

Find the values of the parameters $a, b \in \mathbb{R}$ so that the function $f(x) = a \times \sin(x) + b \times e^{-x}$ satisfies the following conditions simultaneously:

• It passes through the point $(0, 3)$.

• Its derivative at the origin is $f'(0) = 2$.

Given the function $f(x) = \ln(\cos(x))$, find:

1. a) The first derivative $f'(x)$.
2. b) The second derivative $f''(x)$, simplifying the result using standard trigonometric functions.

Calculate the derivative of the function $f(x) = \arctan\left(\frac{x-1}{x+1}\right)$ and simplify the resulting expression. Which simpler function does this derivative correspond to?

Calculate the derivative of the following function, simplifying the final result as much as possible using trigonometric identities:

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

Calculate the derivative of the following composite function and simplify the result as much as possible using trigonometric identities:

What curiosity do you observe in the final simplified result?

Using logarithmic differentiation, find the derivative of the function $f(x) = x^{\sqrt{x}}$ for $x > 0$.

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

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

Given the transcendental function defined by $f(x) = (\arctan x)^x$ for $x > 0$, use the technique of logarithmic differentiation to calculate its derivative function $f'(x)$. Once obtained, calculate the exact value of the gradient of the tangent line to the graph of $f$ at the point with x-coordinate $x = 1$.

To properly size a steel beam subjected to asymmetric torsion, an engineer needs to optimise a stress function given by $E(x) = \frac{\sqrt{x^2 - 1} \times e^{3x}}{(2x + 1)^4}$ for $x > 1$. Apply logarithmic differentiation to obtain the simplified expression of the derivative $E'(x)$.

Determine the values of the real parameters $a$ and $b$ so that the following piecewise function is differentiable on its entire domain:

Explain, with reasoning, the relationship between continuity and differentiability at the breakpoint $x = 1$.

Consider the curve implicitly defined by the equation:

1. a) Find the expression for the derivative $y'$ in terms of $x$ and $y$ using implicit differentiation.
2. b) Find the equation of the normal line to the curve at the point $P(3, 1)$.

Sea la función $f(x) = x^3 + 2x + \cos(x - 1)$.

1. a) Demuestra que $f(x)$ es biyectiva en todo $\mathbb{R}$ (puedes apoyarte en el estudio de su crecimiento).

1. b) Sabiendo que $f(1) = 4$, calcula el valor de la derivada de la función inversa en ese punto, es decir, $(f^{-1})'(4)$.

Find the general expression of the nth derivative ($f^{(n)}(x)$) for the rational function:

Hint: Decompose the function into partial fractions before beginning successive differentiation.

A surveillance drone moves following the trajectory given by the function $f(x) = \sqrt{x^2 + 1}$. An operator located at point $A(2, 0)$ watches the drone.

Calculate the point on the trajectory where the drone is at the minimum distance from the operator.

Note: To minimize the distance $d$, it is sufficient to minimize the square of the distance $d^2$. You will need to differentiate the resulting distance function.

In a biotechnology laboratory, the volume of a bacterial population (in $mm^3$) grows according to the function $V(t) = 20 \times e^{0.5t}$, where $t$ is time in hours. At the same time, the radius of the circular culture dish where the bacteria grow expands such that the area occupied by the population is $A(t) = \pi \times [r(t)]^2$.

Assuming that the thickness of the bacterial layer is constant and equal to $0.1 mm$ (so that $V = A \times h$), calculate the instantaneous rate of change of the radius of the population at time $t = 2$ hours.

A harmonic oscillator in a mechanical system has a position given by $x(t) = e^{-2t} \cos(3t)$. Calculate the velocity $v(t) = x'(t)$ and acceleration $a(t) = x''(t)$, and check whether it satisfies the differential equation $x''(t) + 4x'(t) + 13x(t) = 0$.