Differentiation rules

Apply the product rule to find the derivative of the following functions (do not expand the brackets before differentiating):

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

1. b) $g(x) = (x + 2) \cdot (x^3 - 4x)$

Calculate the derivative of these rational functions using the quotient rule:

1. a) $f(x) = \frac{3x - 5}{x + 2}$

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

1. c) $h(x) = \frac{10}{x^2 + 4}$

A sociological study indicates that the number of users of a new social network (in millions) is given by the function $U(t) = \frac{40t}{t^2 + 4}$, where $t$ is the time in months since its launch. Apply the quotient rule to find the derivative function $U'(t)$ and determine in which month user growth stops and begins to decrease. What is the rate of user loss after one year ($t=12$)?

Apply the chain rule to calculate the derivative function of these composite functions:

1. a) $f(x) = (3x^2 - 5)^4$

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

1. c) $h(x) = (1 - x)^5$

Calculate the equation of the tangent line to the curve $f(x) = \sqrt{x^2 + 5}$ at the point where said curve intersects the line $y = 3$ (consider only the solution in the first quadrant). Once the line is found, determine the area of the triangle formed by this tangent line with the coordinate axes.

The number of views of a viral video (in thousands) is given by the function $V(t) = 2 \cdot e^{0,5t}$, where $t$ is the time in days. Knowing that the derivative of $e^{u}$ is $u' \cdot e^u$, calculate the rate of growth of the views on the second day ($t = 2$).

Calculate the derivative of the following function containing a square root, simplifying the final result:

$f(x) = \sqrt{x^2 + e^x}$

Use the chain rule for functions of the form $\sqrt{u}$.

Obtain the derivative function of the following function using the product rule and the derivative of the exponential function:

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

Factorise the expression for $f'(x)$ by taking out the exponential term as a common factor.