Exercises: Differentiation rules and derivative functions of standard functions

A fireworks rocket takes off vertically from a launch pad. The height $h$ in metres with respect to time $t$ in seconds is given by the polynomial function $h(t) = t^3 - 4t^2 + 5t + 2$. Calculate the derivative function $h'(t)$, which represents the instantaneous velocity of the rocket at any instant $t$.

A social media account has analysed its growth and determined that the number of followers $S$ evolves according to the exponential function $S(t) = 150 \cdot e^{0,4t}$, where $t$ is the elapsed time in months since its creation. Determine the derivative function $S'(t)$ to find the rate of follower growth.

Calculate the derivative function of the following functions by applying the differentiation rules for elementary functions (powers and roots):

1. a) $f(x) = x^7 - 5x^3 + \pi$

1. b) $g(x) = \sqrt{x} + \frac{1}{x^2}$

Given the quadratic function $f(x) = x^2 - 6x + 5$, find:

1. a) The derivative function $f'(x)$.

1. b) The value of the slope of the tangent line to the graph of the function at the point $P$ with abscissa $x = 4$ (marked in the figure).

Find the derivative function of the following transcendental functions using standard derivative tables:

1. a) $f(x) = 4 \cdot \ln(x) + \log_{2}(x)$

1. b) $g(x) = 3^x - 5 \cdot \sin(x)$

A spring oscillates following simple harmonic motion described by the function $y(x) = 5 \cdot \cos(x) + 2$, where $y$ is the position in centimetres and $x$ is time. Obtain the expression for the derivative function $y'(x)$, which represents the oscillation velocity of the spring.

Calculate the derivative of the following simple composite functions by applying the chain rule directly:

1. a) $f(x) = (2x + 3)^5$

1. b) $g(x) = e^{x^2 - 1}$

Given the function $f(x) = \frac{x^4}{4} - \sqrt[3]{x}$, calculate its derivative function $f'(x)$ and simplify the result, writing the root back in radical form.

In a 2nd year Bachillerato exam, Marcos has to calculate the derivative function of $f(x) = \sin^3(2x)$. On his answer sheet, he writes the following:

"Since it is a power, I bring down the exponent and differentiate the sine:

$f'(x) = 3\sin^2(2x) \cdot \cos(2x)$".

1. a) Identify precisely which fundamental step of the chain rule Marcos has omitted when decomposing the layers of the composite function. Explain why his reasoning is erroneous.
2. b) Correct the exercise by writing out the complete differentiation chain step by step, identifying each of the elementary functions that compose $f$ (from the outermost to the innermost), and obtain the correct simplified expression for $f'(x)$.

You want to calculate the derivative function of $f(x) = \ln\left(\frac{e^{3x}}{(x^2+1)^4}\right)$ for all $x \in \mathbb{R}$. Two students propose different strategies:

• Path A (Direct differentiation): Immediately apply the formula $(\ln u)' = \frac{u'}{u}$, previously calculating the derivative of the fraction using the quotient rule and the chain rule for the denominator.

• Path B (Prior simplification using logarithms): Use the properties of logarithms ($\ln(A/B) = \ln A - \ln B$, $\ln(A^k) = k \ln A$, and $\ln(e^u) = u$) to decompose $f(x)$ into a sum and difference of simpler terms before differentiating.

1. a) Compare both paths and justify with reasons which of the two is preferable in terms of lower computational workload and lower probability of making sign or simplification errors.
2. b) Apply the most efficient path to calculate $f'(x)$ and express the result as a single irreducible algebraic fraction.

Calculate the derivative function of the following composite function, applying the chain rule and simplifying the result as much as possible:

$f(x) = \cos^3(e^{x^2+1})$

Given the rational function involving natural logarithms:

$g(x) = \frac{\ln(x^2 + 4)}{x}$

Find the expression of its derivative $g'(x)$ and determine the values of $x$ for which the gradient of the tangent line to the graph of $g(x)$ is equal to zero.

A surveillance drone follows a trajectory whose height in meters is given by the function $h(t) = 15t \cdot e^{-0,2t}$, where $t$ is the time in seconds since takeoff.

1. a) Calculate the function that represents the drone's vertical velocity, $v(t) = h'(t)$.

1. b) What is the drone's vertical velocity at time $t = 5$ seconds?

Find the values of the parameters $a, b \in \mathbb{R}$ so that the function $f(x) = a \cdot \sin(x) + b \cdot e^{-x}$ satisfies the following conditions simultaneously:

• It passes through the point $(0, 3)$.

• Its derivative at the origin is $f'(0) = 2$.

Use logarithmic differentiation to find the derivative of the following power-exponential function:

$f(x) = (x^2 + 1)^{\sin(x)}$

Calculate the equation of the tangent line to the curve $f(x) = \arctan\left(\frac{x}{2}\right)$ at the point with abscissa $x = 2$.

The figure shows the graph of $f$, its horizontal asymptotes and the point $P$ of the curve whose abscissa is $x = 2$.

Given the function $f(x) = \ln(\cos(x))$, find:

1. a) The first derivative $f'(x)$.
2. b) The second derivative $f''(x)$, simplifying the result using standard trigonometric functions.

Calculate the derivative of the following function, simplifying the final result as much as possible using trigonometric identities:

$f(x) = \ln\left( \sqrt{\frac{1 + \sin x}{1 - \sin x}} \right)$

Calculate the derivative of the function $f(x) = \arctan\left(\frac{x-1}{x+1}\right)$ and simplify the resulting expression. Which simpler function does this derivative correspond to?

Using logarithmic differentiation, find the derivative of the function $f(x) = x^{\sqrt{x}}$ for $x > 0$.

At an AEMET weather station in Córdoba during a summer heatwave, the ambient temperature $T(t)$ in degrees Celsius ($^{\circ}\text{C}$) over a 24-hour day fits the model:

$T(t) = 31 - 9 \cdot \cos\left(\frac{\pi \cdot (t - 4)}{12}\right)$,

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

1. a) Determine the derivative function $T'(t)$, clearly stating each of the differentiation rules used (constant multiple rule, chain rule, and trigonometric rule).
2. b) Calculate the exact value and the value rounded to two decimal places of $T'(10)$ (at 10:00 h) and of $T'(16)$ (at 16:00 h).
3. c) Interpret the physical and contextual meaning of the values and signs obtained in the previous section (indicating the units of measurement corresponding to the derivative) and determine at what time of day the temperature rises most rapidly.

Consider the transcendental function $f(x) = \arctan(2x) - x$.

1. a) Without calculating the derivative function analytically, make an estimate of the value of the slope of the tangent line at $x = 0$ by calculating the average rate of change (ARC) over the interval $[0;\, 0,1]$ using a calculator (in radian mode). From the result, do you expect the function to be increasing or decreasing in an immediate neighbourhood of $x = 0$?
2. b) Apply the differentiation rules to obtain the simplified algebraic expression for the derivative function $f'(x)$.
3. c) Evaluate the exact value of $f'(0)$, compare it with the estimated value obtained in section (a), and validate with sound reasoning whether your previous hypothesis about the local monotonicity of the function is confirmed.

Given the transcendental function defined by $f(x) = (\arctan x)^x$ for $x > 0$, use the technique of logarithmic differentiation to calculate its derivative function $f'(x)$. Once obtained, calculate the exact value of the gradient of the tangent line to the graph of $f$ at the point with x-coordinate $x = 1$.

Determine the values of the real parameters $a$ and $b$ so that the following piecewise function is differentiable on its entire domain:

Explain, with reasoning, the relationship between continuity and differentiability at the breakpoint $x = 1$.

Find the general expression of the nth derivative ($f^{(n)}(x)$) for the rational function:

Hint: Decompose the function into partial fractions before beginning successive differentiation.

Consider the curve implicitly defined by the equation:

1. a) Find the expression for the derivative $y'$ in terms of $x$ and $y$ using implicit differentiation.
2. b) Find the equation of the normal line to the curve at the point $P(3, 1)$.

Sea la función $f(x) = x^3 + 2x + \cos(x - 1)$.

1. a) Demuestra que $f(x)$ es biyectiva en todo $\mathbb{R}$ (puedes apoyarte en el estudio de su crecimiento).

1. b) Sabiendo que $f(1) = 4$, calcula el valor de la derivada de la función inversa en ese punto, es decir, $(f^{-1})'(4)$.

A surveillance drone moves following the trajectory given by the function $f(x) = \sqrt{x^2 + 1}$. An operator located at point $A(2, 0)$ watches the drone.

Calculate the point on the trajectory where the drone is at the minimum distance from the operator.

Note: To minimize the distance $d$, it is sufficient to minimize the square of the distance $d^2$. You will need to differentiate the resulting distance function.

Calculate the derivative of the following composite function and simplify the result as much as possible using trigonometric identities:

What curiosity do you observe in the final simplified result?

In a biotechnology laboratory, the volume of a bacterial population (in $mm^3$) grows according to the function $V(t) = 20 \cdot e^{0,5t}$, where $t$ is time in hours. At the same time, the radius of the circular culture dish where the bacteria grow expands such that the area occupied by the population is $A(t) = \pi \cdot [r(t)]^2$.

Assuming that the thickness of the bacterial layer is constant and equal to $0,1 mm$ (so that $V = A \cdot h$), calculate the instantaneous rate of change of the radius of the population at time $t = 2$ hours.

To properly size a steel beam subjected to asymmetric torsion, an engineer needs to optimise a stress function given by $E(x) = \frac{\sqrt{x^2 - 1} \cdot e^{3x}}{(2x + 1)^4}$ for $x > 1$. Apply logarithmic differentiation to obtain the simplified expression of the derivative $E'(x)$.

A harmonic oscillator in a mechanical system has a position given by $x(t) = e^{-2t} \cos(3t)$. Calculate the velocity $v(t) = x'(t)$ and acceleration $a(t) = x''(t)$, and check whether it satisfies the differential equation $x''(t) + 4x'(t) + 13x(t) = 0$.

An unknown function $f(x)$, differentiable and strictly positive on the interval $(0, +\infty)$, satisfies that its logarithmic derivative is given by the expression:

$\frac{f'(x)}{f(x)} = \frac{2\ln x + 1}{x}$.

Furthermore, it is known that the graph of $f$ passes through the point $(1, 1)$.

1. a) Justify why the expression $\frac{f'(x)}{f(x)}$ formally coincides with the derivative of the composite function $\ln(f(x))$.
2. b) Reasonably deduce the expression for the function $g(x) = \ln(f(x))$ by identifying a function whose derivative is $\frac{2\ln x + 1}{x}$. To do so, decompose the quotient into the sum of two fractions and recognise them as derivatives of elementary functions or powers of $\ln x$.
3. c) From the condition $f(1) = 1$, find the corresponding constant and solve for the original function $f(x)$, expressing it in the power-exponential form $u(x)^{v(x)}$.
4. d) Check your result by directly differentiating the obtained function $f(x)$ using logarithmic differentiation and verifying that it accurately reproduces the starting quotient.

Given the function $f(x) = \ln\sqrt{\frac{1 - \cos(2x)}{1 + \cos(2x)}}$, defined on the open interval $I = \left(0, \frac{\pi}{2}\right)$:

Two solution methods are considered to calculate its derivative function $f'(x)$:

• Method 1 (Algebraic properties of logarithms): First rewrite $f(x) = \frac{1}{2}\big(\ln(1 - \cos(2x)) - \ln(1 + \cos(2x))\big)$ and differentiate as a difference of natural logarithms applying the chain rule.

• Method 2 (Prior trigonometric simplification): Transform the radicand using the double-angle identities ($1 - \cos(2x) = 2\sin^2 x$ and $1 + \cos(2x) = 2\cos^2 x$) before applying any differentiation rule.

1. a) Fully carry out Method 2 to find $f'(x)$ in its most compact expression, written as a quotient in terms of a single double-angle trigonometric ratio. Justify why the sign inside the square root generates no ambiguity on the interval $I$.
2. b) Carry out Method 1 and show in detail all the algebraic and trigonometric steps required to simplify the resulting sum of fractions until obtaining exactly the same expression as in Method 2.
3. c) Critically compare both methods: analyse which one minimizes the probability of algebraic error and conclude what general criterion of prior simplification should be adopted when facing this type of derivatives in university entrance examinations (PAU).