Exercises: Definition of the derivative of a function at a point

Using the definition of derivative by means of the limit $f'(a) = \lim_{h \to 0} \frac{f(a+h) - f(a)}{h}$, calculate the value of $f'(2)$ for the function $f(x) = x^2 + 3$.

Find the instantaneous rate of change (derivative) of the function $f(x) = 3x - 5$ at the point with x-coordinate $x = 1$, applying the limit of the difference quotient.

The distance covered by an object in free fall on a fictitious planet is given by $s(t) = 4t^2$, where $s$ is measured in meters and $t$ in seconds. Calculate the instantaneous velocity at $t = 3$ s applying the definition of derivative as a limit.

Given the function $f(x) = 2x^2 - x$, use the alternative expression of the definition $f'(a) = \lim_{x \to a} \frac{f(x) - f(a)}{x - a}$ to find the derivative at the point $a = -1$.

Prove, by explicitly calculating the one-sided limits of the difference quotient, that the absolute value function $f(x) = |x - 3|$ is not differentiable at the point $x = 3$.

A cyclist in a velodrome covers a distance given by the function $d(t) = t^2 + 5t$ (measured in meters, with $t$ in seconds). Calculate their instantaneous speed at $t = 2$ seconds using strictly the definition of derivative by limits.

Find the equation of the tangent line to the curve $f(x) = x^2$ at the point with x-coordinate $x = 2$. To do this, first calculate the gradient $m = f'(2)$ using the definition of derivative by means of the limit.

Calculate the derivative of the constant function $f(x) = 7$ at the point $x = 5$ applying the formal definition with the limit. Briefly explain the geometrical meaning of the result obtained.

Using the definition of derivative by means of limits, calculate $f'(3)$ for the rational function $f(x) = \frac{1}{x - 1}$.

Find the derivative of the radical function $f(x) = \sqrt{x + 5}$ at the point $x = 4$, applying the definition of derivative as a limit and multiplying by the conjugate to resolve the indeterminate form.

In a technology component factory, the total cost of producing $q$ units is given by the function $C(q) = 2\,000 + 40q + 2q^2$ in pesos. Determine the marginal cost (the derivative of the cost function) when exactly $q = 20$ units are produced, by solving the limit of the formal definition of derivative.

Determine whether the following piecewise function is differentiable at $x = 2$, first checking its continuity and then applying the definition of one-sided derivatives:

$f(x) = \begin{cases} x^2 & \text{if } x \le 2 \\ 4x - 4 & \text{if } x > 2 \end{cases}$

Find the equation of the tangent line to the function $f(x) = \frac{x}{x + 2}$ at the origin. You must calculate the gradient by evaluating the limit of the definition of derivative at $x = 0$.

The altitude of an agricultural observation drone follows the equation $h(t) = -5t^2 + 20t + 2$, with height $h$ expressed in meters and time $t$ in seconds. Using the definition of derivative by means of the limit for a general point $t$, obtain the velocity function $v(t) = h'(t)$ and determine at what exact instant the drone reaches its maximum height (when its velocity is zero).

The attached graph shows a continuous function that has a sharp corner. Define analytically an absolute value function that corresponds to this graph and prove rigorously, by calculating the one-sided limits of the difference quotient, why the function is not differentiable at $x = 2$.

Calculate, using the definition of derivative with the limit for a general point $x$, the derivative function $f'(x)$ of $f(x) = x^3 - 3x$. Then, use the result obtained to find the points on the curve where the tangent line is completely horizontal.

Determine the exact values of the parameters $a$ and $b$ so that the piecewise function

$f(x) = \begin{cases} ax^2 + b & \text{if } x < 1 \\ \frac{1}{x} & \text{if } x \ge 1 \end{cases}$

is differentiable at $x = 1$. You must justify your answer by applying the definition of continuity and evaluating the limits of the one-sided derivatives using the difference quotient.

Calculate the derivative of the function $f(x) = \sqrt{x^2 + 3}$ at the point $x = 1$ using exclusively the definition of derivative as a limit. Show all the algebraic steps necessary to resolve the indeterminate form.

Suppose $f(x)$ is an even function (that is, it satisfies $f(-x) = f(x)$ for all real numbers $x$) and is known to be differentiable at the point with x-coordinate $x = 0$. Prove analytically, using the definition of derivative by means of the limit $\lim_{h \to 0} \frac{f(h) - f(0)}{h}$, that $f'(0) = 0$ must necessarily hold.

The temperature $T$ (measured in degrees Celsius) of a computer processor as a function of time $t$ (in minutes) after starting an intensive calculation process is given by the expression $T(t) = \frac{4t}{t^2 + 1} + 30$. Calculate the instantaneous rate of heating (the derivative) at exactly $t = 2$ minutes, evaluating step by step the limit of the definition of derivative.

Find the equations of all the tangent lines to the curve $f(x) = x^3 - x$ that are strictly parallel to the line with equation $y = 2x + 1$. To solve this, you must first obtain the general expression for the gradient $f'(x)$ applying the definition of derivative by limits.

Given the function $f(x) = x|x|$, prove using the definition of derivative by calculating the one-sided limits of the difference quotient that the function is smooth (differentiable) at $x = 0$, and find the exact value of $f'(0)$.

Using the general definition of derivative $f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$, prove step by step that the derivative function of $f(x) = \frac{1}{\sqrt{x}}$ is exactly $f'(x) = -\frac{1}{2x\sqrt{x}}$ for any value of $x > 0$.

The quarterly profit of a renewable energy company as a function of its investment $x$ (in millions of pesos) follows a quadratic model of the form $B(x) = -ax^2 + bx$. Given that maximum profit is reached with an exact investment of $10$ million pesos and that this maximum profit amounts to $500$ million, deduce the values of the constants $a$ and $b$. Subsequently, prove using the definition of derivative by limits that the instantaneous rate of change at that point satisfies $B'(10) = 0$.