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 $f(t) = \cos(5t) - t^3$.

Aoife has to calculate the derivative function of $h(x) = (3x^2 - 5)^4$. In her exam, she writes the following:

And from there, she deduces that $h'(1) = 4(3\times 1^2 - 5)^3 = 4(-2)^3 = -32$.

1. a) Accurately identify which differentiation rule she applied incompletely and explain in your own words which factor she forgot to add according to the chain rule, as well as what that factor conceptually represents.
2. b) Correct Aoife's procedure, write the correct expression for the derivative function $h'(x)$, and calculate the true value of $h'(1)$.

The task is to calculate the derivative function of $f(x) = \frac{6x^4 - 9x^2}{3x^2}$ for all $x \ne 0$.

• Method 1: Directly apply the quotient rule of differentiation: $\left(\frac{u}{v}\right)' = \frac{u'v - uv'}{v^2}$.

• Method 2: First divide each term in the numerator by the monomial in the denominator to rewrite $f(x)$ as a simple polynomial, and then differentiate term by term.

1. a) Carry out the calculation of the derivative by applying both methods step by step.
2. b) Compare both approaches: which one turns out to be faster and less prone to calculation or sign errors? Justify why Method 2 is only advisable when the denominator is a single monomial.

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 metres. 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 decimetres 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.

A photovoltaic solar panel installed in a house in Galway undergoes slight thermal expansion during the middle hours of the day. Its width in metres varies according to the function $a(t) = 1 + 0.02t$, and its length according to $l(t) = 2 + 0.05t$, where $t$ is the elapsed time in hours since 11:00 ($t \ge 0$). The collection surface area is given by $S(t) = a(t) \times l(t)$.

1. a) Before differentiating, give a reasoned estimate of whether the instantaneous rate of increase of the area at $t = 0$ ($S'(0)$) will be greater than, less than, or equal to the sum of the rates at which its sides grow ($a'(0) + l'(0) = 0.02 + 0.05 = 0.07\text{ m}^2/\text{h}$). Base your conjecture on the geometric interpretation of the product rule (the growth of a rectangle distributed into two strips).
2. b) Apply the product rule to calculate the derivative function $S'(t)$ and evaluate the exact value of $S'(0)$.
3. c) Check whether your initial estimate was correct and explain how the initial dimensions ($a(0) = 1\text{ m}$ and $l(0) = 2\text{ m}$) play a role in the overall growth rate of the surface area.

During a period of continuous autumn rainfall, the volume of water stored in a reservoir in the Shannon river basin evolves over time according to the model:

where $V(t)$ represents the additional volume stored in cubic hectometres ($\text{hm}^3$) and $t$ is the elapsed time in days since the start of the storm ($t \ge 0$).

1. a) Apply the quotient rule to obtain the derivative function $V'(t)$ and simplify the numerator of the resulting fraction as much as possible.
2. b) Determine on which day $t$ the instantaneous filling rate vanishes ($V'(t) = 0$). Analyse the sign of $V'(t)$ before and after that day and interpret what physically happens to the reservoir's volume.
3. c) Calculate the rate of change of the volume at $t = 2$ days and at $t = 6$ days. State the units in which this derivative is measured and interpret the physical meaning of its sign at both moments.

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 $L'(x) = \frac{2}{\cos(2x)}$.

It is known that a function $h(x)$ is defined by the product $h(x) = x^2 \times g(x)$, where $g(x)$ is a differentiable function on all of $\mathbb{R}$ whose algebraic expression is not directly known. However, it has been experimentally verified that the tangent line to the graph of $h(x)$ at the point with abscissa $x = 3$ has the equation $y = 12x - 9$.

1. a) From the equation of the tangent line, deduce in a justified manner the exact numerical values of $h(3)$ and the derivative $h'(3)$.
2. b) Apply the product rule to the expression $h(x) = x^2 \times g(x)$ to establish an algebraic relationship between $h'(3)$, $g(3)$, and $g'(3)$. Solve for and calculate the exact numerical value of $g(3)$ and $g'(3)$.
3. c) From the sign obtained for $g'(3)$, reason whether the function $g(x)$ is increasing or decreasing in the vicinity of $x = 3$.
4. d) Using the deduced results, write the equation of the tangent line to the graph of $y = g(x)$ at the point with abscissa $x = 3$.

In the notes, it is established that a quotient is equivalent to the product $\frac{f(x)}{g(x)} = f(x) \times [g(x)]^{-1}$, so that its derivative arises from combining the product rule and the chain rule.

1. a) Carry out this derivation formally: differentiate the expression $f(x) \times [g(x)]^{-1}$ by applying the product rule and the chain rule to $[g(x)]^{-1}$, factor out $[g(x)]^{-2}$, and show that the standard formula is obtained exactly: $\frac{f'(x)g(x) - f(x)g'(x)}{[g(x)]^2}$.
2. b) Generalise the reasoning: show that the derivative of a function of the form $H(x) = \frac{f(x)}{[g(x)]^n}$ (where $n \in \mathbb{N}$) can be directly expressed as:

1. c) Apply the formula derived in part b to find the derivative of $k(x) = \frac{2x - 1}{(x^2 + 3)^3}$.
2. d) Choose and justify: compare this procedure with using the traditional quotient rule on $k(x)$ and reason why the generalised formula avoids cumbersome simplifications of repeated factors in the numerator and denominator.