Exercises: Derivatives of standard functions

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 product rule to find the derivative of the function:

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

Find the derivative of the following quotient of functions:

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

Calculate the derivative of the following trigonometric function:

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

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

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

Find the derivative of the following composite exponential function:

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

Calculate the derivative of the function involving a radical:

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

Determine the derivative of the following composite logarithmic function:

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

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.

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 centimeters and $t$ in seconds. Determine the velocity function $v(t)$ of the oscillator by differentiating the position function.

The total production cost, in pesos, of $q$ units of a high-precision electronic component is modeled by the function $C(q) = \frac{4\,000q^2 + 10\,000}{q+2}$. Calculate the marginal cost $C'(q)$ when exactly $q = 10$ units are produced.

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

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

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 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.

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

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

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 $f'(x) = \tan x \times \sec x$.

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 traveled 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 ($\sin \theta_1 = \sin \theta_2$).

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 $I''(t) = 0$).

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

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