Derivative of a Function at a Point
A business doesn't need to know "the cost in general": it needs to know what happens right at the production level it is at today. How much does it cost to produce one more item right now? That question is answered by the derivative evaluated at a specific point, $a$. In this section you will learn to compute it, interpret it, and use it to draw tangent lines. At the end we will come back to that question.
From average change to instantaneous change
If $f$ describes a quantity (followers, cost, revenue), the average rate of change between $x=a$ and $x=b$ is
$$\frac{\Delta y}{\Delta x}=\frac{f(b)-f(a)}{b-a}$$and it is the slope of the secant line joining the points $(a,f(a))$ and $(b,f(b))$. It tells you how much $f$ changes, on average, for each unit of $x$ over that stretch.
If the endpoint $b$ gets closer to $a$, the secant rotates until it becomes the tangent, and the average rate of change becomes the instantaneous rate of change: the derivative $f'(a)$.
A content creator has observed that the number of followers of their channel during the first week follows the function $f(t) = t^2 + 2t + 100$, where $t$ is time in…
A content creator has observed that the number of followers of their channel during the first week follows the function $f(t) = t^2 + 2t + 100$, where $t$ is time in days.
Calculate the Average Rate of Change (AROC) of the number of followers between the second day ($t = 2$) and the fifth day ($t = 5$). What does this value mean in the context of the problem?
The definition of the derivative at a point
This follows the four-step rule: compute $f(a+h)$, then $f(a+h)-f(a)$, divide by $h$, and take the limit as $h\to0$. Direct substitution gives $\frac{0}{0}$; you have to simplify first.
When there is a square root, the $\frac00$ indeterminate form is removed by multiplying by the conjugate.
Given the polynomial function $f(x) = x^2 - 4x + 5$, calculate the value of its derivative at the point $x = 3$ using the definition of derivative using limits: $$f'(a)…
Given the polynomial function $f(x) = x^2 - 4x + 5$, calculate the value of its derivative at the point $x = 3$ using the definition of derivative using limits:
$$f'(a) = \lim_{h \to 0} \frac{f(a+h) - f(a)}{h}$$The number of downloads of a new trending app, in thousands, is given by the function $D(t) = 2t^2 - t$, where $t$ is the time in hours since its launch. Calculate the…
The number of downloads of a new trending app, in thousands, is given by the function $D(t) = 2t^2 - t$, where $t$ is the time in hours since its launch.
Calculate the instantaneous download speed (the derivative) at $t = 4$ hours using the limit:
$$D'(4) = \lim_{x \to 4} \frac{D(x) - D(4)}{x - 4}$$Given the rational function $f(x) = \frac{2}{x}$, calculate $f'(2)$ by applying the definition of derivative at a point. Explain whether the function is increasing or…
Given the rational function $f(x) = \frac{2}{x}$, calculate $f'(2)$ by applying the definition of derivative at a point. Explain whether the function is increasing or decreasing at that point based on the sign of the result.
Use the definition of derivative at a point by calculating the limit: $$f'(a) = \lim_{h \to 0} \frac{f(a+h) - f(a)}{h}$$ to find the value of the derivative of the…
Use the definition of derivative at a point by calculating the limit:
$$f'(a) = \lim_{h \to 0} \frac{f(a+h) - f(a)}{h}$$to find the value of the derivative of the function $f(x) = \frac{3}{x-2}$ at the point with x-coordinate $x = 4$. Explain the geometric meaning of the result obtained in relation to the gradient of the curve at that point.
Calculate, using exclusively the definition of the derivative via the limit of the difference quotient ($h \to 0$), the value of $f'(4)$ for the function: $$f(x) =…
Calculate, using exclusively the definition of the derivative via the limit of the difference quotient ($h \to 0$), the value of $f'(4)$ for the function:
$$f(x) = \sqrt{2x + 1}$$Justify each step of the process, especially the elimination of the indeterminate form.
Geometric meaning: the tangent line
$f'(a)$ is the slope of the tangent line to the graph at the point $(a,f(a))$. If $f'(a)>0$ the function is increasing at that point; if $f'(a)<0$ it is decreasing; if $f'(a)=0$ the tangent is horizontal.
If the problem gives you a graph with the tangent drawn, the slope is read as $\dfrac{\text{vertical change}}{\text{horizontal change}}$ between two points on that line.
Here, if the horizontal change is 2 units and the vertical change rises 1, the slope is $\frac12$, that is, $f'(a)=\frac12$ at point $P$.
To differentiate without using the limit every time, these rules are enough for this section (they are studied in detail in "Differentiation rules"):
| Function | Derivative |
|---|---|
| $c$ (constant) | $0$ |
| $x^n$ | $n\,x^{n-1}$ |
| $u+v$ | $u'+v'$ |
| $\dfrac{u}{v}$ | $\dfrac{u'v-uv'}{v^2}$ |
| $\ln x$ | $\dfrac1x$ |
| $e^{u}$ | $u'\,e^{u}$ |
Observe the following graph of the function $f(x)$ and the tangent line to it at point $P$. From the visual information, state with reasoning what the value of the…
Observe the following graph of the function $f(x)$ and the tangent line to it at point $P$.
From the visual information, state with reasoning what the value of the derivative of the function is at the point with x-coordinate $x = 2$, that is, find $f'(2)$.
Find the equation of the tangent line to the graph of the function $f(x) = 3x^2 - 5$ at the point with x-coordinate $x = 1$. Note: Remember that the equation of the…
Find the equation of the tangent line to the graph of the function $f(x) = 3x^2 - 5$ at the point with x-coordinate $x = 1$.
Note: Remember that the equation of the tangent line is given by $y - f(a) = f'(a)(x - a)$.
Find the equation of the tangent line to the graph of the function $f(x) = x^2 - 5x + 2$ at the point where the function crosses the y-axis (the $Y$ axis).
Find the equation of the tangent line to the graph of the function $f(x) = x^2 - 5x + 2$ at the point where the function crosses the y-axis (the $Y$ axis).
Rate of change and marginal cost
The derivative estimates the change for a small increment, but it is not exact: the tangent is a straight line and the curve moves away from it.
Go back to the beginning: how much it costs to produce one more item right now is the marginal cost $C'(a)$ at the current production level.
A courier company has found that the cost of shipping a package (in pesos) depends on its weight $x$ (in kg) according to the function $C(x) = \sqrt{4x + 9}$. Calculate,…
A courier company has found that the cost of shipping a package (in pesos) depends on its weight $x$ (in kg) according to the function $C(x) = \sqrt{4x + 9}$. Calculate, using the definition of the derivative as a limit, the instantaneous rate of change of the cost for a package weighing exactly $4$ kg and interpret the result in economic terms (marginal cost).
The number of users of a new social network (in thousands) evolves according to the function $U(t) = \frac{20t}{t+2}$, where $t$ is the time in months since its launch.…
The number of users of a new social network (in thousands) evolves according to the function $U(t) = \frac{20t}{t+2}$, where $t$ is the time in months since its launch. Find:
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The average rate of change of users between month $t=1$ and month $t=3$.
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The instantaneous growth rate (derivative) in month $t=2$ using differentiation rules and compare both values.
The revenue function of a tech startup is given by $I(x) = \frac{8\,000x}{x+40}$, where $x$ is the marketing investment in thousands of pesos and $I(x)$ is the revenue…
The revenue function of a tech startup is given by $I(x) = \frac{8\,000x}{x+40}$, where $x$ is the marketing investment in thousands of pesos and $I(x)$ is the revenue in thousands of pesos.
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Calculate the Average Rate of Change (AROC) in the interval $[40; 50]$.
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Calculate the Instantaneous Rate of Change (derivative) at $x = 40$.
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What error is made if we use the derivative at $x=40$ to estimate the revenue increase when going from 40 to 50 thousand pesos of investment?
Points with a given slope
Sometimes you are not given the point but a condition on the tangent (horizontal, parallel to a line, perpendicular, an angle, passing through another point). You have to find the points.
When the data are about unknown parameters, each condition gives an equation: "passes through a point" gives an equation involving $f$; "tangent with slope $m$ at $x=a$" or "zero derivative" gives one involving $f'$. For example, the slope of the tangent of a cubic is $f'(x)$ (a parabola), and its minimum is at the vertex of that parabola.
Determine the points on the graph of the function $f(x) = x^2 - 6x$ at which the tangent line is horizontal. Hint: A line is horizontal when its slope is equal to zero.
Determine the points on the graph of the function $f(x) = x^2 - 6x$ at which the tangent line is horizontal.
Hint: A line is horizontal when its slope is equal to zero.
Determine the points on the graph of the function $f(x) = \frac{1}{3}x^3 - x^2 - 2x + 5$ where the tangent line is parallel to the line $y = 1 - 2x$.
Determine the points on the graph of the function $f(x) = \frac{1}{3}x^3 - x^2 - 2x + 5$ where the tangent line is parallel to the line $y = 1 - 2x$.
Consider the function $f(x) = \frac{k}{x^2+1}$. Determine the value of the parameter $k \neq 0$ so that the tangent line to the curve at the point with x-coordinate…
Consider the function $f(x) = \frac{k}{x^2+1}$. Determine the value of the parameter $k \neq 0$ so that the tangent line to the curve at the point with x-coordinate $x = 1$ makes an angle of $135^\circ$ with the positive direction of the $OX$-axis. Once $k$ is found, write the equation of that tangent line.
The trajectory of a delivery drone follows the curve $f(x) = x^2 - 6x + 8$ (in hectometers). The drone operator wants to program a straight-line approach manoeuvre from…
The trajectory of a delivery drone follows the curve $f(x) = x^2 - 6x + 8$ (in hectometers). The drone operator wants to program a straight-line approach manoeuvre from a point $P(a, f(a))$ on the trajectory, such that the tangent line to the curve at that point passes exactly through the charging station, located at the origin $(0,0)$.
Find the coordinates of the possible points of tangency and the equations of these lines.
Given the function $f(x) = x^3 + ax^2 + bx + c$, determine the values of the parameters $a, b$ and $c$ knowing that: 1. The graph of $f(x)$ passes through the point…
Given the function $f(x) = x^3 + ax^2 + bx + c$, determine the values of the parameters $a, b$ and $c$ knowing that:
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The graph of $f(x)$ passes through the point $(0, 5)$.
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The gradient of the tangent line at $x = 0$ is minimum.
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There is a stationary point (zero derivative) at $x = 2$.
Extension: differentiability and one-sided derivatives
The one-sided derivatives are the limits of the difference quotient from the left ($h\to0^-$) and from the right ($h\to0^+$). When the function changes formula at $x=a$ (piecewise rates, for example), you have to compute both.
Continuity does not imply differentiability, but differentiability does imply continuity.
Parameters that make a piecewise function differentiable
You impose both conditions: continuity (the values match) and equal one-sided derivatives. That gives two equations for two parameters.
If a piece includes $\ln x$ or $\frac{a}{x}$, the idea is the same: set the values and the derivatives equal at the junction point. Remember $(\ln x)'=\frac1x$ and $\left(\frac{a}{x}\right)'=-\frac{a}{x^2}$.
Absolute value
With absolute value, $|g(x)|$ is written piecewise according to the sign of $g$. In $f(x)=|x^2-4|$, for $x<2$ (near 2) we have $x^2-4<0$, so $f(x)=4-x^2$ and $f'(2^-)=-4$; for $x>2$, $f(x)=x^2-4$ and $f'(2^+)=4$. Since they are different, there is a corner point at $x=2$.
Consider the piecewise function: $$f(x) = \begin{cases} 2x + 1 & \text{if } x < 2 \\ x^2 + 1 & \text{if } x \geq 2 \end{cases}$$ Study the differentiability of the…
Consider the piecewise function:
$$f(x) = \begin{cases} 2x + 1 & \text{if } x < 2 \\ x^2 + 1 & \text{if } x \geq 2 \end{cases}$$Study the differentiability of the function at the point $x = 2$ by checking whether the one-sided derivatives exist and are equal. Is the function continuous at that point?
Given the piecewise function: $$f(x) = \begin{cases} x^2 - a & \text{if } x < 2 \\ bx + 1 & \text{if } x \geq 2 \end{cases}$$ Calculate the values of $a$ and $b$ so that…
Given the piecewise function:
$$f(x) = \begin{cases} x^2 - a & \text{if } x < 2 \\ bx + 1 & \text{if } x \geq 2 \end{cases}$$Calculate the values of $a$ and $b$ so that the function is differentiable at the point $x = 2$. Remember that for a function to be differentiable at a point, it must first be continuous at that point.
Study the differentiability of the function $f(x) = |x^2 - 4|$ at the point $x = 2$ by calculating its one-sided derivatives. Interpret graphically why the function is…
Study the differentiability of the function $f(x) = |x^2 - 4|$ at the point $x = 2$ by calculating its one-sided derivatives. Interpret graphically why the function is not differentiable at that x-value.
Study the differentiability of the function $f(x) = x \times |x - 2|$ at the point $x = 2$. To do this: 1. Express the function as a piecewise function. 2. Calculate the…
Study the differentiability of the function $f(x) = x \times |x - 2|$ at the point $x = 2$. To do this:
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Express the function as a piecewise function.
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Calculate the one-sided derivatives at $x = 2$ using the definition of derivative.
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Interpret graphically whether the point $(2, f(2))$ is a "sharp point" or a point of "smooth transition".
A logistics company has modelled the maintenance cost of its fleet of electric vans using a piecewise function $f(x)$. To ensure that there are no abrupt changes in…
A logistics company has modelled the maintenance cost of its fleet of electric vans using a piecewise function $f(x)$. To ensure that there are no abrupt changes in operating costs, the function must be differentiable across its entire domain. Determine the values of $a$ and $b$ so that the function is differentiable at $x = 1$:
$$f(x) = \begin{cases} ax^2 + bx + 2 & \text{if } x \le 1 \\ \ln(x) + \frac{a}{x} & \text{if } x > 1 \end{cases}$$Extension: the normal line
Find the equation of the normal line to the curve $f(x) = e^{x-1} + 2x$ at the point with x-coordinate $x = 1$. Remember that the normal line is perpendicular to the…
Find the equation of the normal line to the curve $f(x) = e^{x-1} + 2x$ at the point with x-coordinate $x = 1$. Remember that the normal line is perpendicular to the tangent line at the point of contact.
Find the equation of the normal line to the graph of the function $f(x) = e^{2x-4}$ at the point where the graph intersects the horizontal line $y = 1$. Remember that…
Find the equation of the normal line to the graph of the function $f(x) = e^{2x-4}$ at the point where the graph intersects the horizontal line $y = 1$. Remember that the normal line is perpendicular to the tangent at the point of contact.