Exercises: Differentiation rules

Calculate the derivative function of $f(x) = 5x^4 - \frac{3}{x^2} + 2\sqrt{x} - 7$, applying the basic power rules of differentiation and linearity.

Determine the derivative of the function $g(t) = (3t^2 - 5t) \times e^t$ using the product rule.

Apply the quotient rule to find the derivative of the rational function $h(x) = \frac{2x + 1}{x^2 - 3}$.

Calculate the derivative of the composite function $f(x) = \sin(4x^3 - x)$, applying the chain rule correctly.

Find the first derivative of the logarithmic function $y = \ln(x^2 + 5x + 6)$.

Calculate the derivative of the function $f(x) = 3^{2x} + \arctan(x)$, recalling the differentiation rule for exponentials with a base other than $e$ and for inverse trigonometric functions.

Determine the second derivative, $f''(t)$, of the function $f(t) = \cos(5t) - t^3$.

The number of daily streams of a viral song evolves according to the function $R(t) = 50 t^2 e^{-0.2t}$, where $t$ is measured in days since its release and $R(t)$ in thousands of streams. Calculate the instantaneous rate of change $R'(t)$ by applying differentiation rules and state its fully factorised expression.

The average production cost of a high-precision electronic component is given by $C(x) = \frac{\ln(x + 1)}{x}$, where $x$ is the number of batches manufactured ($x > 0$). Determine the function that describes the rate of change of the average cost, $C'(x)$.

The temperature of an industrial oven after being switched off drops according to the cooling law $T(t) = 20 + 80 \cos^2\left(\frac{\pi t}{4}\right)$, with $T$ in degrees Celsius and $t$ in hours. Find the expression for the cooling rate $T'(t)$ using the chain rule.

The profile of the roof of an Olympic stadium follows the plane curve given by $f(x) = \frac{100}{x^2 + 25}$, in meters. Calculate the general equation of the tangent line at the point with x-coordinate $x = 5$ to design a straight structural support that rests tangentially at that point.

An articulated robotic arm moves in such a way that the angle of its main joint (in radians), as a function of a control parameter $x$, is $\theta(x) = \arcsin\left(\frac{x-1}{x+1}\right)$ for $x > 1$. Obtain and state the fully simplified form of $\theta'(x)$.

The position of a pneumatic piston inside a cylinder is described by the equation of motion $s(t) = \ln(1 + t^2)$, where $s$ is in decimeters and $t$ in seconds. Calculate the expression for the acceleration of the piston, corresponding to the second derivative $s''(t)$, and determine at what time $t > 0$ this acceleration becomes zero.

The trajectory of a satellite during a complex orbital maneuver is described in a reference plane by the curve implicitly defined by the equation $x^2 y - 3y^3 x = 10$. Calculate the value of the slope of the trajectory, $y'$, at the point with coordinates $(-2, 1)$.

Let $f(x)$ and $g(x)$ be two functions whose graphs are shown in the figure. The graph of $f$ is the parabola $f(x)=-x^2+6x-3$, which passes through the points $(2,\,5)$ and $(4,\,5)$. The graph of $g$ is the line that passes through the points $(2,\,4)$ and $(4,\,2)$. The composite function $h(x)=f(g(x))$ and the product function $p(x)=f(x)\times g(x)$ are defined.

Calculate the exact values of:

1. a) $h'(2)$

1. b) $p'(4)$

The intensity of an electromagnetic signal attenuated as it passes through a dense medium follows the function $I(x) = x e^{-x}$. Calculate the first three derivatives $I'(x)$, $I''(x)$, and $I'''(x)$, and from the results obtained, deduce and write down the general formula for the nth derivative $I^{(n)}(x)$.

In an AC RLC circuit, the charge on the capacitor is given by the function $q(t) = e^{-\alpha t} (A \sin(\omega t) + B \cos(\omega t))$. Show, by calculating $q'(t)$ and $q''(t)$ in detail, that for the constants $A=1$, $B=0$, $\alpha=2$, and $\omega=3$, the function satisfies the second-order differential equation:

The sound pressure level in an anechoic chamber, after applying a frequency filter, is given by the function:

Show, using the laws of logarithms before differentiating and applying trigonometric identities in the final result, that $L'(x) = \frac{2}{\cos(2x)}$.