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 metres 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 $a = -1$.

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.

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 $x = 3$.

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

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

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.

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 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.

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 metres 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).

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.