Differentiation rules

Calculate the derivative function of the following basic polynomial functions using the power rule and the sum rule:

1. a) $f(x) = 5x^3 - 4x^2 + 7x - 2$

1. b) $g(x) = \frac{x^4}{2} - 3x^2 + 10$

1. c) $h(x) = -x^5 + \frac{2x^3}{3} + x$

To differentiate functions with roots, we must first transform them into powers. Express the following functions as powers of $x$ and then calculate their derivative:

1. a) $f(x) = 4\sqrt{x}$

1. b) $g(x) = \sqrt[3]{x^2}$

1. c) $h(x) = \frac{2}{\sqrt{x}}$

An online clothing store has found that its daily profits in pesos follow the function $B(t) = -t^2 + 20t + 150$, where $t$ is the time in hours since the server was opened. Calculate the function $B'(t)$, which represents the rate at which profits change, and find its value when 4 hours have passed ($t = 4$).

Find the slope of the tangent line to the graph of the function $f(x) = x^2 - 6x + 8$ at the point with abscissa $x = 1$. The figure shows the graph of $f$ (blue) and the tangent line (orange) at the point $P$ with abscissa $x = 1$. Is the function increasing or decreasing at that point? Justify your answer based on the sign of the derivative.

Find the equation of the tangent line to the graph of the function $f(x) = x^2 - 6x + 5$ at the point with abscissa $x = 4$. To do so, first obtain the slope using the derivative and then use the point-slope form.

The figure shows the parabola and the point on its graph whose abscissa is $x = 4$.

Determine the points on the curve $f(x) = x^3 - 3x^2 - 7x + 10$ at which the tangent line is parallel to the line $y = 2x - 5$.

Hint: Remember that two lines are parallel if they have the same gradient.

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

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

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

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

Calculate the derivative function of the following rational function using the quotient and power rules:

$f(x) = \dfrac{3x^2 - 5x}{2x + 4}$

Simplify the numerator of the resulting expression as much as possible.

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

Given the function $g(x) = (x^2 - k)^3$, determine the value or values of the real parameter $k$ knowing that the second derivative of the function at the point $x=1$ is equal to $0$, that is, $g''(1) = 0$. Take special care when applying the chain rule repeatedly.

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 $V(t) = 2 \times e^{0.5t}$, where $t$ is the time in days. Knowing that the derivative of $e^{u}$ is $u' \times e^u$, calculate the rate of growth of the views on the second day ($t = 2$).

Apply the chain rule to calculate the derivative of the following composite function, which combines a natural logarithm with a polynomial:

$f(x) = \ln(x^3 - 4x^2 + 8)$

State the domain of the resulting derivative function.

Given the function $f(x) = ax^2 + bx + 3$, determine the values of parameters $a$ and $b$, knowing that the derivative of the function at $x = 1$ is equal to $7$ ($f'(1) = 7$) and that the derivative at $x = 0$ is equal to $-2$ ($f'(0) = -2$).

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:

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

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

Calculate the derivative of the following composite function, simplifying the final result as much as possible:

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

Apply the laws of logarithms before differentiating to simplify the calculation and use the chain rule for exponential functions.

A telephone company has determined that its monthly profits (in thousands of pesos) follow the function $B(x) = -x^2 + 48x - 150$, where $x$ is the number of hundreds of new customers acquired. The figure shows the graph of $B$ and the point corresponding to $x = 20$.

Calculate the instantaneous rate of change of the profit (marginal profit) when $2000$ customers have been acquired (that is, for $x = 20$). Interpret whether the profit is increasing or decreasing at that moment.

A courier company has found that the daily fuel cost of its fleet (in pesos) depends on the average speed $v$ (in km/h) according to the function $C(v) = 10v + \frac{40\,000}{v}$. Using differentiation rules, approximate the rate of change of the cost using the derivative when the speed increases from $80$ km/h to $81$ km/h. At what speed does the cost stop decreasing and start increasing?

It is known that the profit function of a tech company follows the expression $B(x) = ax^2 + \frac{b}{x}$, where $x$ is the investment in advertising in thousands of pesos. Determine the values of parameters $a$ and $b$, knowing that when the investment is $1$ thousand pesos ($x=1$), the profit is $5$ thousand pesos and, moreover, at that same point, the instantaneous rate of change of profit is $1$ thousand pesos for every thousand pesos invested.

Given the piecewise function:

$f(x) = \begin{cases} ax^2 + bx + 1 & \text{if } x < 1 \\ \ln(x) + 2 & \text{if } x \ge 1 \end{cases}$

Determine the values of $a$ and $b$ so that the function is differentiable across its entire domain. To do so, remember that differentiability necessarily implies continuity at the jump point.

Find the points on the graph of the function $f(x) = \frac{x-1}{x+1}$ at which the tangent line is parallel to the line $y = \frac{1}{2}x - 3$. State the equations of these tangent lines.