Differentiation rules and derivatives of standard functions

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).

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.

A fireworks rocket takes off vertically from a launch pad. The height $h$ in metres 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 $t$.

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 $x = 2$.

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.

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

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

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 $x = 1$.