Differentiation rules

Calculate the derivative of the following polynomial functions using the basic rules of differentiation:

1. a) $f(x) = 5x^3 - 2x^2 + 7x - 10$

1. b) $g(x) = \frac{1}{4}x^4 + \frac{2}{3}x^3 - 5$

1. c) $h(x) = x^5 - \sqrt{2}x$

A content creator has observed that the number of views of one of their videos on a social network follows the function $f(t) = t^2 + 20t + 100$, where $t$ is the time in hours since it was published.

Calculate the Average Rate of Change (A.R.O.C.) of the number of views in the interval $[1, 4]$. Interpret the result obtained in the context of the problem.

Given the function $f(x) = (3x - 5) \cdot e^x$, find its derivative by applying the product rule. Simplify the result by factorising where possible.

A cosmetics company has estimated that the daily profit (in hundreds of euros) generated by the sale of a new facial serum follows the function $B(t) = 20t \cdot e^{-0,2t}$, where $t$ is the time in months since its launch. The figure shows the graph of $B(t)$ and marks the point corresponding to $t=5$.

1. a) Obtain the expression of the function $B'(t)$, which represents the rate of change of the profit.

1. b) Calculate the value of the derivative 5 months after its launch and interpret the result obtained in economic terms.

Calculate the derivative of the following rational function using the quotient rule:

State for which values of $x$ the derivative of this function does not exist, based on its domain.

Consider the function $f(x) = \dfrac{e^{2x}}{x^2 + 1}$.

1. a) Calculate its derivative $f'(x)$, simplifying the expression as much as possible.

1. b) Determine whether there is any point where the curve has a horizontal tangent (zero gradient). If so, state the coordinates of that point.

Given the function $f(x) = \dfrac{x^2 + k}{x - 1}$, calculate the value of the constant $k$ knowing that the gradient of the tangent line to the graph of $f(x)$ at the point with abscissa $x = 2$ is equal to $-3$. Once $k$ has been found, write the complete equation of this tangent line.

Apply the chain rule to find the derivative of the following composite functions:

1. a) $f(x) = (2x^2 + 1)^4$

1. b) $g(x) = \ln(5x - 2)$

1. c) $h(x) = e^{3x^2}$

Given the function $f(x) = \sqrt{x^2 + 9}$, calculate its derivative using the chain rule. Once obtained, find the value of the gradient of the tangent line to the curve at the point $x = 4$.

Consider the curve $y = x^3 - 6x^2 + 4$ and the line $r: y = -9x + 2$.

1. a) Find the points of the curve at which the tangent line is parallel to the line $r$.

1. b) Write the equations of the tangent lines to the curve at those points.

Given the function $f(x) = x^2 \cdot \ln(x)$, calculate:

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

1. b) The second derivative $f''(x)$.

1. c) The equation of the normal line to the graph at the point with abscissa $x = e$. (Remember that the normal line is perpendicular to the tangent at the point of contact).

Apply the technique of logarithmic differentiation to find the expression for the derivative of:

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

Calculate the derivative function of the following expression using logarithmic differentiation. Take into account that the base and the exponent are functions of $x$:

Show all intermediate steps, applying the laws of logarithms correctly before differentiating.

Given the rational function $f(x) = \frac{1}{2x - 3}$, obtain a general expression for its nth derivative, $f^{(n)}(x)$.

To do this, calculate the 1st, 2nd, 3rd, and 4th order derivatives, identify the pattern in numbers and signs, and write it in terms of $n$ and the factorial $n!$.