Exercises: Differentiation rules and derivative functions of standard functions

A surveillance drone takes off vertically from a platform. The height $h$ in feet with respect to time $t$ in seconds is given by the polynomial function $h(t) = t^3 - 4t^2 + 5t + 2$. Find the derivative function $h'(t)$, which represents the instantaneous velocity of the drone at any time $t$.

A social media account has analyzed 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 time elapsed in months since its creation. Determine the derivative function $S'(t)$ to find the growth rate of followers.

Find the derivative of each of the following functions using differentiation rules for elementary functions (powers and roots):

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

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

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

1. a) Find the derivative $f'(x)$.

1. b) Find the slope of the tangent line to the graph of the function at the point with x-coordinate $x = 4$.

Find the derivative of the following transcendental functions using standard derivative rules:

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 inches and $x$ is the time. Find the expression for the derivative function $y'(x)$, which represents the velocity of oscillation of the spring.

Find the derivative of the following simple composite functions using the chain rule directly:

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

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

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

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

A surveillance drone follows a trajectory whose height in feet is given by the function $h(t) = 15t \cdot e^{-0.2t}$, where $t$ is the time in seconds since takeoff.

1. a) Find the function that represents the vertical velocity of the drone, $v(t) = h'(t)$.
2. b) What is the vertical velocity of the drone at $t = 5$ seconds?

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

• Passes through the point $(0, 3)$.

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

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

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

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

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.

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

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

Given the transcendental function defined by $f(x) = (\arctan x)^x$ for $x > 0$, use the technique of logarithmic differentiation to find its derivative $f'(x)$. Once obtained, calculate the exact value of the slope 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 throughout its domain:

Explain with reasoning the relationship between continuity and differentiability at the transition point $x = 1$.

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

Hint: Decompose the function using partial fractions before taking successive derivatives.

Consider the curve defined implicitly 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 $P(3, 1)$.

Let the function be $f(x) = x^3 + 2x + \cos(x - 1)$.

1. a) Prove that $f(x)$ is bijective on all of $\mathbb{R}$ (you can rely on the study of its increasing behavior).
2. b) Knowing that $f(1) = 4$, calculate the value of the derivative of the inverse function at that point, that is, $(f^{-1})'(4)$.

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

Find 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 do you notice about the final simplified result?

In a biotechnology laboratory, the volume of a population of bacteria (in $mm^3$) grows according to the function $V(t) = 20 \cdot e^{0.5t}$, where $t$ is the time in hours. At the same time, the radius of the circular culture plate where the bacteria grow expands so that the area occupied by the population is $A(t) = \pi \cdot [r(t)]^2$.

If we assume that the thickness of the bacterial layer is constant and equal to $0.1 mm$ (so that $V = A \cdot h$), find the instantaneous rate of change of the radius of the population at $t = 2$ hours.