The Derivative Function
A Metrobús bus leaves the station and its position changes over time. At every instant the speedometer shows a different number. Is there a function that gives you that number at any moment? Yes: it is the derivative function, and on this page you learn how to obtain it and how to read what it says about the original function.
The derivative function
Until now you calculated $f'(a)$ at one specific point $a$. If you set up the same limit for any $x$, the result is no longer a number, but another function.
Apply the definition of the derivative using the limit calculation $f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$ to find the derivative function of…
Apply the definition of the derivative using the limit calculation $f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$ to find the derivative function of $f(x) = 3x^2 - 5x + 2$.
Finding f'(x) with the rules
Using the limit every time is long. With the rules you see in detail in other sections, you can differentiate directly. Here you use them as a tool:
| Function | Derivative |
|---|---|
| $k$ (constant) | $0$ |
| $x^n$ (with $n$ an integer, positive or negative, or a fraction) | $n x^{n-1}$ |
| $e^x$ | $e^x$ |
| $a^x$ | $a^x \ln a$ |
| $\ln x$ | $\dfrac{1}{x}$ |
| $\sin x$ | $\cos x$ |
| $\cos x$ | $-\sin x$ |
| Rule | Formula |
|---|---|
| Constant multiple | $(c\,f)'=c\,f'$ |
| Sum | $(f+g)'=f'+g'$ |
| Product | $(fg)'=f'g+fg'$ |
| Quotient | $\left(\dfrac{f}{g}\right)'=\dfrac{f'g-fg'}{g^2}$ |
| Chain | $\big(f(g(x))\big)'=f'(g(x))\cdot g'(x)$ |
The chain rule makes sense like this: if $g$ changes at a certain rate and $f$ responds to $g$ at another rate, the total rate is the product of both.
The multiplying constant "stays" and only the part with $x$ is differentiated. And the formula for $x^n$ also works with negative exponents: first rewrite $\dfrac1{x^2}=x^{-2}$.
Calculate the derivative function of the following polynomial and exponential function, applying the basic rules of differentiation:…
Calculate the derivative function of the following polynomial and exponential function, applying the basic rules of differentiation:
$f(x) = \frac{3}{4}x^4 - 2x^3 + \frac{5}{x^2} - 3e^x$
Obtain the derivative function of $f(x)$ applying the product rule. Simplify the result as much as possible. $f(x) = (2x^3 - x + 4) \times \ln(x)$
Obtain the derivative function of $f(x)$ applying the product rule. Simplify the result as much as possible.
$f(x) = (2x^3 - x + 4) \times \ln(x)$
Find the derivative function applying the quotient rule. Express the final result with the numerator simplified. $f(x) = \frac{4x - 3}{x^2 + 5}$
Find the derivative function applying the quotient rule. Express the final result with the numerator simplified.
$f(x) = \frac{4x - 3}{x^2 + 5}$
Use the chain rule to determine the derivative function of the following composite trigonometric function: $f(x) = \cos(5x^3 - 2x^2 + 1)$
Use the chain rule to determine the derivative function of the following composite trigonometric function:
$f(x) = \cos(5x^3 - 2x^2 + 1)$
Calculate the exact value of the derivative of the function $f(x) = \sqrt{3x^2 + 4}$ evaluated at the point with x-coordinate $x = 2$.
Calculate the exact value of the derivative of the function $f(x) = \sqrt{3x^2 + 4}$ evaluated at the point with x-coordinate $x = 2$.
Apply the properties of logarithms to simplify the expression for $f(x)$ before differentiating, and then calculate its simplified derivative function:…
Apply the properties of logarithms to simplify the expression for $f(x)$ before differentiating, and then calculate its simplified derivative function:
$f(x) = \ln\left( \sqrt{\frac{1 + \sin(x)}{1 - \sin(x)}} \right)$
The cumulative number of views of a viral video $t$ hours after its publication follows an exponential growth law given by $R(t) = 1500 \times 3^{0.4t}$. Obtain the…
The cumulative number of views of a viral video $t$ hours after its publication follows an exponential growth law given by $R(t) = 1500 \times 3^{0.4t}$.
Obtain the function describing the instantaneous growth rate of views, $R'(t)$, and calculate this rate after $5$ hours.
Domain of differentiability
When a piecewise function has parameters, two conditions are required at the joining point: that it be continuous (both pieces have the same value) and that the one-sided derivatives be equal. Two conditions, two equations for two parameters.
Determine the values of the real parameters $a$ and $b$ such that the following piecewise-defined function is differentiable at $x = 2$:…
Determine the values of the real parameters $a$ and $b$ such that the following piecewise-defined function is differentiable at $x = 2$:
$f(x) = \begin{cases} ax^2 + 3x & \text{if } x \le 2 \\ bx^3 - 4 & \text{if } x > 2 \end{cases}$
Consider the function defined using a modulus $f(x) = |x^2 - 4|$. Express $f(x)$ as a piecewise function and calculate its derivative function $f'(x)$. Prove…
Consider the function defined using a modulus $f(x) = |x^2 - 4|$.
Express $f(x)$ as a piecewise function and calculate its derivative function $f'(x)$. Prove analytically why the function is not differentiable at the corner points indicated on the graph.
The sign of f'(x) and the growth of f
Observe the graph of the function $f(x)$ representing the trajectory of a projectile as a function of time. From the visual representation, determine the sign of the…
Observe the graph of the function $f(x)$ representing the trajectory of a projectile as a function of time.
From the visual representation, determine the sign of the derivative function $f'(x)$ at the x-coordinates $x = 1$, $x = 2$, and $x = 3$. Justify your answer based on the geometric concept of the derivative.
The following graph represents the derivative function $f'(x)$ of a continuous function $f(x)$ across all of $\mathbb{R}$. Observing exclusively the behavior of $f'(x)$,…
The following graph represents the derivative function $f'(x)$ of a continuous function $f(x)$ across all of $\mathbb{R}$.
Observing exclusively the behavior of $f'(x)$, determine the intervals where the original function $f(x)$ is increasing and where it is decreasing. At which x-coordinates will $f(x)$ have relative extrema?
The following panel shows two curves, A (blue) and B (green). It is known that one of them corresponds to a polynomial function $f(x)$ and the other to its derivative…
The following panel shows two curves, A (blue) and B (green). It is known that one of them corresponds to a polynomial function $f(x)$ and the other to its derivative function $f'(x)$.
Give reasons to identify which curve represents $f(x)$ and which represents $f'(x)$ based on the geometric relationship between the intercepts, the relative extrema, and the intervals of increase/decrease. Next, deduce the sign of the second derivative $f''(x)$ at the x-coordinate $x = 0$.
Critical points and local extrema
Vertical asymptote and parameters
Some exercises give a rational function with a piece of information such as "it has a vertical asymptote at $x=-2$". A vertical asymptote at $x=a$ appears when the denominator is zero at $a$ but the numerator is not: as you approach $a$ the values of $f$ grow or decrease without bound ($\lim_{x\to a}f(x)=\pm\infty$), and the graph hugs the line $x=a$ without touching it.
In a chemical kinetics experiment, the concentration of a reactant as a function of time $t$ (in minutes) fits the expression $C(t) = \frac{8t}{t^2 + 4}$ in moles per…
In a chemical kinetics experiment, the concentration of a reactant as a function of time $t$ (in minutes) fits the expression $C(t) = \frac{8t}{t^2 + 4}$ in moles per liter.
Find the derivative function $C'(t)$ that represents the reaction rate. At what time $t > 0$ is the reaction rate zero?
Find the x-coordinates of the points on the curve $f(x) = e^{3x}(x^2 - 4x + 2)$ at which the tangent line is horizontal.
Find the x-coordinates of the points on the curve $f(x) = e^{3x}(x^2 - 4x + 2)$ at which the tangent line is horizontal.
Given the rational function $f(x) = \frac{ax^2 + b}{cx + 1}$, determine the values of the real parameters $a, b, c$ knowing that its graph passes through the point…
Given the rational function $f(x) = \frac{ax^2 + b}{cx + 1}$, determine the values of the real parameters $a, b, c$ knowing that its graph passes through the point $(0, 3)$, has a vertical asymptote at $x = -2$, and has a horizontal tangent line at the point with x-coordinate $x = 1$.
The derivative as a rate of change and as a slope
When $f$ describes a quantity that changes, $f'$ is its instantaneous rate of change. With time it is velocity; with costs it is marginal cost. On a graph, $f'(a)$ is the slope of the tangent line at $x=a$.
The equation of motion of an autonomous test vehicle is given by the function $s(t) = 2t^3 - 15t^2 + 24t + 10$, where $s$ is the position in meters and $t$ is the time…
The equation of motion of an autonomous test vehicle is given by the function $s(t) = 2t^3 - 15t^2 + 24t + 10$, where $s$ is the position in meters and $t$ is the time in seconds ($t \ge 0$).
Find the velocity function $v(t)$ and the acceleration function $a(t)$. Determine at what times the vehicle comes to a momentary standstill.
The total production cost, in pesos, of $x$ units of a high-precision electronic component is modeled by the function $C(x) = 0.40x^3 - 12x^2 + 800x + 10\,000$.…
The total production cost, in pesos, of $x$ units of a high-precision electronic component is modeled by the function $C(x) = 0.40x^3 - 12x^2 + 800x + 10\,000$.
Calculate the marginal cost function $C'(x)$ and evaluate its value when exactly $x = 15$ units are produced. Interpret the result obtained in the context of the problem.
In the design of a suspension bridge, the profile of the main cable takes the form of a catenary whose equation in a coordinate system is…
In the design of a suspension bridge, the profile of the main cable takes the form of a catenary whose equation in a coordinate system is $f(x) = \frac{e^{x/10} + e^{-x/10}}{2}$ (with $x$ and $y$ in meters).
Calculate the angle of inclination $\alpha$ that the tangent line to the cable makes with the horizontal at the anchorage point located at $x = 10$ meters.
The second derivative and successive derivatives
For $f''$ to exist at the joining point of a piecewise function, three things are required at that point: $f$ continuous, $f'$ continuous (the one-sided derivatives of $f$ coincide), and the one-sided second derivatives equal. That is, the same procedure as before, applied one level further.
Determine the second derivative, $f''(x)$, of the function $f(x) = x^4 \times \ln(x)$.
Determine the second derivative, $f''(x)$, of the function $f(x) = x^4 \times \ln(x)$.
Find the general expression for the nth derivative, $f^n(x)$, of the function $f(x) = \ln(3x - 2)$. Hint: calculate the first four successive derivatives and deduce the…
Find the general expression for the nth derivative, $f^n(x)$, of the function $f(x) = \ln(3x - 2)$. Hint: calculate the first four successive derivatives and deduce the pattern.
Show that the function $y = e^{-2x} \times \sin(3x)$ is a particular solution to the second-order linear differential equation: $y'' + 4y' + 13y = 0$
Show that the function $y = e^{-2x} \times \sin(3x)$ is a particular solution to the second-order linear differential equation:
$y'' + 4y' + 13y = 0$
Consider the function $f(x) = \begin{cases} e^{ax} - 1 & \text{if } x \le 0 \\ b \sin(x) + c x^2 & \text{if } x > 0 \end{cases}$. Determine the mathematical relationship…
Consider the function $f(x) = \begin{cases} e^{ax} - 1 & \text{if } x \le 0 \\ b \sin(x) + c x^2 & \text{if } x > 0 \end{cases}$.
Determine the mathematical relationship that must exist between the parameters $a, b, c$ for the second derivative to exist at the origin, that is, for $f''(0)$ to be a well-defined real value.