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

An online clothing store has found that its daily profits in euros 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 ($t = 4$).

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

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

Cian has to differentiate the function $f(x) = (2x - 5) \times e^x$ in Mathematics class. In his notebook, he writes the following resolution:

"Since the derivative of $2x - 5$ is $2$ and the derivative of $e^x$ is $e^x$, the derivative of the product is the product of the derivatives: $f'(x) = 2 \times e^x$".

Identify and explain in detail the conceptual error made by Cian (supporting your answer with the product rule explained in the notes) and write down the correct resolution by calculating and simplifying $f'(x)$ by factoring out common factors.

Consider the function $f(x) = (x^2 - 1) \times e^x$.

1. a) Without calculating the derivative, estimate with reasoning the sign of the instantaneous rate of change $f'(2)$. To do so, qualitatively analyze how both factors, $u(x) = x^2 - 1$ and $v(x) = e^x$, behave when $x$ increases slightly starting from $x = 2$: are both factors positive at that point? Are both increasing? What consequence does this have on the growth rate of the area of the rectangle they represent, according to the justification of the product rule?
2. b) Formally calculate the derivative function $f'(x)$ by applying the product rule, factor out $e^x$, and evaluate $f'(2)$. Check whether the numerical value obtained confirms your prediction from the previous part.

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

To calculate the derivative function of $f(x) = (4x - 3)^2$, two 5th Year students propose different approaches:

• Aoife suggests directly applying the chain rule: differentiate the outer power keeping the parentheses intact and multiply by the derivative of the inside.

• Cian suggests first expanding the squared binomial to obtain a polynomial and then differentiating term by term using the power and sum rules.

1. a) Apply both methods separately, showing all intermediate steps, and verify that the same expression is obtained.
2. b) Justify which of the two strategies is more advantageous if the function to differentiate were $g(x) = (4x - 3)^{10}$ instead of a square power.

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.

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.

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.

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.

A financial analysis consulting firm models the growth of the yield of a pension plan using a function with the structure $f(x) = (ax + b) \times e^{3x}$, where $a$ and $b$ are constant real parameters and $x$ is the time in years elapsed since its opening.

1. a) Apply the product rule and the chain rule to obtain the general algebraic expression of $f'(x)$, leaving the exponential function $e^{3x}$ factored out.
2. b) Knowing that the econometric study requires the derivative function to be exactly $f'(x) = (12x + 19) \times e^{3x}$, deduce the unique values of $a$ and $b$ using the method of undetermined coefficients.
3. c) Reason in detail whether it would be possible to find values of $a$ and $b$ such that the derivative $f'(x)$ is reduced to an ordinary polynomial (that is, without the presence of the factor $e^{3x}$). Justify your answer based on the algebraic properties of the derivative of the exponential function.

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.

A telephone company has determined that its monthly profits (in thousands of euros) 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.