Differentiation rules and derivatives of standard functions

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:

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

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

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 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.

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

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 $x = 2$.

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?

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.

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 that represents the drone's vertical velocity, $v(t) = h'(t)$.

1. b) What is the drone's vertical velocity at time $t = 5$ seconds?

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 $x''(t) + 4x'(t) + 13x(t) = 0$.

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

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 $x = 1$.

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.

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 $P(3, 1)$.

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 $x = 1$.

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.