Definition of the derivative of a function at a point
In 4th ESO you saw that, as you narrow the interval $[a, a+h]$ until $h$ tends to $0$, the average rate of change settles down to a single number. Now it is time to write this with the full rigour of the limit: that is the derivative, and in 1st Bachillerato you will use it as a calculation tool, not just as an intuitive idea.
The limit of the difference quotient
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…
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.
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.
Two ways of writing the same definition
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…
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 $f(x) = \frac{1}{x - 1}$.
Using the definition of derivative by means of limits, calculate $f'(3)$ for the rational function $f(x) = \frac{1}{x - 1}$.
One-sided derivatives and corner points
Just as with limits, if the difference quotient is calculated only with $h\to0^+$ or $h\to0^-$, you obtain the one-sided derivatives. The function is differentiable at $a$ only if both exist and are equal.
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…
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:…
Determine whether the following piecewise function is differentiable at $x = 2$, first checking its continuity and then applying the definition of one-sided derivatives:
$f(x) = \begin{cases} x^2 & \text{if } x \le 2 \\ 4x - 4 & \text{if } x > 2 \end{cases}$
Differentiability and continuity
Determine whether the following piecewise function is differentiable at $x = 2$, first checking its continuity and then applying the definition of one-sided derivatives:…
Determine whether the following piecewise function is differentiable at $x = 2$, first checking its continuity and then applying the definition of one-sided derivatives:
$f(x) = \begin{cases} x^2 & \text{if } x \le 2 \\ 4x - 4 & \text{if } x > 2 \end{cases}$
Using the definition to find the tangent line
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…
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…
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 $x = 0$.
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…
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.