Derivatives of standard functions

Find the derivative of the following composite exponential function:

$q(x) = e^{4x^3 - 7x}$

Determine the derivative of the following composite logarithmic function:

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

Aoife has to find the derivative function of $f(x) = 3^x + \ln(4x)$ and on her exam she writes the following:

1. a) Identify the two errors made by Aoife. Explain in detail why it is incorrect to treat $3^x$ as if it were a power function of the type $x^n$ and what she omitted when applying the chain rule to the logarithmic term.
2. b) Provide the correct step-by-step analytical development and calculate the exact value of the slope of the tangent line at $x = 1$.

To compute the derivative of the function $f(x) = \ln\left(\frac{e^{2x}}{x^2 + 1}\right)$, two classmates discuss which approach to take:

• Strategy A (direct): Apply the general formula $(\ln u)' = \frac{u'}{u}$, differentiating the fraction directly using the quotient rule.

• Strategy B (prior simplification): First apply the properties of logarithms $(\ln(A/B) = \ln A - \ln B$ and $\ln(e^k) = k)$ to rewrite $f(x)$ before differentiating term by term.

1. a) Develop the derivative using Strategy B and express the result as a single simplified fraction.
2. b) Justify which of the two strategies is more suitable for minimizing the risk of algebraic errors when differentiating functions with logarithms.

The number of active users of an emerging social network, in thousands, as a function of time $t$ (in months) is given by the expression $U(t) = 50 e^{0.2t^2 - 0.1t}$. Calculate the instantaneous rate of growth of the number of users at time $t = 3$ months.

Use the laws of logarithms to simplify the expression before differentiating, and find the derivative of the function:

$g(x) = \ln\left(\sqrt{\frac{x-2}{x+2}}\right)$

Find the general expression for the nth derivative (of order $n$) of the function $f(x) = \ln(2x - 1)$.

Calculate the derivative of the following trigonometric function:

$f(x) = 5\sin x - 3\cos x + \tan x$

Repeatedly apply the chain rule to find the derivative of the following trigonometric function:

$f(x) = \sin^3(2x^2 + \pi)$

From records of Met Éireann during a heatwave in Kilkenny, the temperature $T$ (in degrees Celsius) throughout a summer day is modeled according to the function:

where $t$ is the time in hours elapsed since midnight ($t \in [0, 24]$).

1. a) Obtain the expression for the instantaneous rate of change of temperature, $T'(t)$, clearly stating the units in which it is measured.
2. b) Calculate and interpret the meaning of $T'(10)$ and $T'(16)$. At which of these two instants is the temperature changing faster?
3. c) Solve the equation $T'(t) = 0$ in the interval $[0, 24]$ with reasoning, and determine at what times of day the absolute minimum and maximum temperatures are recorded, stating the value of both temperatures.

Find the derivative of the following polynomial function:

$f(x) = 4x^5 - \frac{3}{2}x^4 + 7x^2 - \sqrt{2}x + 5$

Apply the chain rule to differentiate the following power of a function:

$p(x) = (3x^2 - 5x + 1)^4$

Calculate the derivative of the function involving a radical:

$r(x) = \sqrt{5x^2 - 3x + 2}$

A light ray travels from the point $A(0, a)$ to the point $B(d, b)$, reflecting off a flat mirror situated on the x-axis at the point $P(x, 0)$. The total distance travelled by the light is given by the function $L(x) = \sqrt{x^2 + a^2} + \sqrt{(d-x)^2 + b^2}$. Calculate $L'(x)$ and show that the condition for minimum distance $L'(x) = 0$ implies that the angle of incidence is equal to the angle of reflection ($\sin \theta_1 = \sin \theta_2$).

Calculate the derivative of the following function that combines inverse trigonometric functions with exponentials and polynomials:

$h(x) = \arctan(e^x) + \arcsin(2x)$

Consider the function $f(x) = \arctan(3x)$.

1. a) Before calculating any derivative, make the following qualitative estimates by reasoning about the graph and the definition of the arctangent:

• What sign do you expect the slope of the tangent line to have for any value of $x$?

• At which of the two points, $x = 0$ or $x = 2$, do you foresee the tangent line having a greater steepness (greater slope)?

1. b) Formally calculate the derivative function $f'(x)$ and find the exact values of $f'(0)$ and $f'(2)$.
2. c) Check whether the obtained values match your estimates and explain what happens to the value of the slope when $x$ becomes infinitely large ($x \to +\infty$), interpreting the result geometrically.

Two students approach calculating the derivative of the function $f(x) = \arcsin(x) + \arccos(x)$, defined on $[-1, 1]$, in different ways:

• Aoife mechanically applies the derivative table rules for both standard functions and adds the algebraic results.

• Seán recalls the geometric property of complementary angles in a right triangle, according to which $\arcsin(x) + \arccos(x) = \frac{\pi}{2}$ for all $x \in [-1, 1]$, and differentiates this constant.

1. a) Develop Aoife's analytical procedure to calculate $f'(x)$ on $(-1, 1)$. What does any observer directly deduce from its derivative about the graph of $f(x)$?
2. b) Explain how Seán's geometric reasoning confirms Aoife's result and compare the efficiency of both approaches.
3. c) Choose and justify the most suitable method to calculate the derivative function of $g(x) = \arctan(x) + \arctan\left(\frac{1}{x}\right)$ for all $x > 0$. Carry out the calculations and explain what geometric property the value of $g'(x)$ reveals about the function $g$.

Apply the product rule to find the derivative of the function:

$g(x) = x^3 \ln x$

Find the derivative of the following quotient of functions:

$h(x) = \frac{e^x}{x^2+1}$

The position of a mass attached to a damped spring is given by the function $x(t) = 5 e^{-0.4t} \cos(3t)$, where $x$ is measured in centimetres and $t$ in seconds. Determine the velocity function $v(t)$ of the oscillator by differentiating the position function.

The total production cost, in euros, of $q$ units of a high-precision electronic component is modelled by the function $C(q) = \frac{200q^2 + 500}{q+2}$. Calculate the marginal cost $C'(q)$ when exactly $q = 10$ units are produced.

Find the equation of the tangent line to the curve defined by $f(x) = x e^{-2x} + \cos(\pi x)$ at the point with x-coordinate $x = 0$.

The temperature $T$ (in degrees Celsius) of an industrial engine as a function of time $t$ (in minutes) after being switched off follows Newton's law of cooling, given by $T(t) = 25 + 85 e^{-0.05t}$. Calculate and interpret the value of $T'(10)$, that is, the rate at which the temperature decreases after 10 minutes.

Determine the values of the real parameters $a$ and $b$ such that the function $f(x) = \frac{ax+b}{x^2+4}$ has a local extremum at the point $P(2, 1)$.

Show, using the quotient rule for differentiation and the fundamental trigonometric identity ($\sin^2 x + \cos^2 x = 1$), that the derivative of the secant function $f(x) = \frac{1}{\cos x}$ is $f'(x) = \tan x \times \sec x$.

In an epidemiological model, the proportion of the population infected by a virus over time $t$ (in weeks) follows a logistic curve given by $I(t) = \frac{1}{1 + 49e^{-0.5t}}$. Calculate the rate of change of the infection $I'(t)$ and determine in which week the disease spreads most rapidly (that is, the point of inflection where $I''(t) = 0$).

Determine the equation of the tangent line to the curve $f(x) = e^x \cos x$ that is parallel to the line $y = -e^{\pi} x + 5$, given that the point of tangency lies in the interval $[0, 2\pi]$.

The monthly profit of a software company, in thousands of euros, is modelled by the function $B(x) = x^2 \ln\left(\frac{10}{x}\right)$, where $x$ is the investment in marketing in thousands of euros ($0 < x \le 10$). Calculate the optimal investment $x$ that maximises the profit and show that it is a local maximum.

In a biomedical study, the blood concentration $C(t)$ (in mg/l) of an active ingredient after being administered orally is modeled using a function from the family:

where $t$ is the time elapsed in hours and the constants $a$ and $k$ are positive real parameters.

1. a) Determine the exact values of $a$ and $k$ knowing that the initial rate of absorption into the bloodstream is $6\text{ mg/(l}\times\text{h)}$ (that is, $C'(0) = 6$) and that the maximum concentration is reached exactly 2 hours after ingestion (that is, $C'(2) = 0$).
2. b) With the values found, calculate the rate of change 4 hours after intake ($C'(4)$) and interpret the sign obtained.
3. c) Prove analytically that for any time $t > 2$ the derivative function strictly satisfies $C'(t) < 0$, and reason what clinical implication this fact has on the evolution of the drug.