The derivative function. Successive derivatives
You already know how to differentiate a function at a point. But $f'(x)$ is not just a single number for each $x$: it is itself a complete function, with its own expression and its own graph. And because it is a function, it can be differentiated again. This opens two doors that you will use throughout the unit: reading a firm's marginal cost, revenue or profit directly from $f'$, and differentiating several times to study the shape of a curve precisely.
From the derivative at a point to the derivative function
Given the function $f(x) = 3x^2 - 5x + 2$, carry out the following tasks: a) Calculate the derivative function $f'(x)$ by applying the differentiation rules. b) Find the…
Given the function $f(x) = 3x^2 - 5x + 2$, carry out the following tasks:
- a) Calculate the derivative function $f'(x)$ by applying the differentiation rules.
- b) Find the value of the slope of the tangent line to the graph of $f(x)$ at the point with abscissa $x = 2$.
- c) Determine the equation of the tangent line at that point.
Economic reading: the marginal function
In economics, the derivative of a cost, revenue or profit function is called a marginal function: it approximates how much the total changes when one more unit is produced or sold. This section is the gateway to the whole unit on economic optimisation.
A phone case manufacturing company has determined that the total cost (in euros) of producing $x$ units is given by the function: $$C(x) = 0,5x^2 + 10x + 200$$ Calculate…
A phone case manufacturing company has determined that the total cost (in euros) of producing $x$ units is given by the function:
$$C(x) = 0,5x^2 + 10x + 200$$Calculate the marginal cost function, $C'(x)$, and determine the approximate cost of producing the 21st unit (that is, calculate $C'(20)$).
Successive derivatives
Calculate the first three derivatives of the polynomial function $f(x) = x^4 - 2x^3 + 5x - 7$. Once obtained, check whether $f'''(1) = 18$ holds.
Calculate the first three derivatives of the polynomial function $f(x) = x^4 - 2x^3 + 5x - 7$. Once obtained, check whether $f'''(1) = 18$ holds.
Given the polynomial function $f(x) = \frac{1}{12}x^4 - \frac{2}{3}x^3 + \frac{5}{2}x^2 - 10$, perform the following operations: a) Calculate the first derivative…
Given the polynomial function $f(x) = \frac{1}{12}x^4 - \frac{2}{3}x^3 + \frac{5}{2}x^2 - 10$, perform the following operations:
- a) Calculate the first derivative $f'(x)$ and the second derivative $f''(x)$.
- b) Determine the values of $x$ for which the second derivative is zero ($f''(x) = 0$).
- c) Calculate the higher-order derivative $f^{(5)}(x)$ and justify the result obtained without needing to carry out all intermediate steps.
The pattern of the $n$th derivative
For simple functions (powers, exponentials, $1/x$), the successive derivatives follow a regular pattern that can be written all at once with the formula for $f^{(n)}(x)$, without differentiating one by one up to the required order.
Given the function $h(x) = \ln(x^2 + 1)$, whose graph appears in the figure, calculate: a) The slope of the tangent line to the curve at the point with abscissa $x = 1$.…
Given the function $h(x) = \ln(x^2 + 1)$, whose graph appears in the figure, calculate:
- a) The slope of the tangent line to the curve at the point with abscissa $x = 1$.
- b) The equation of the second derivative function $h''(x)$.
- c) The points of the graph where the second derivative is equal to zero.
Determine the general expression for the $n$-th derivative of the rational function $f(x) = \dfrac{1}{3x - 2}$, defined for $x \neq \dfrac{2}{3}$. To do so, calculate…
Determine the general expression for the $n$-th derivative of the rational function $f(x) = \dfrac{1}{3x - 2}$, defined for $x \neq \dfrac{2}{3}$. To do so, calculate the first four successive derivatives, identify the recurring pattern, and use factorial notation $n!$ and powers to write $f^{(n)}(x)$.