Derivative of a Polynomial
An app on your phone records how many users it has each month, and you want to know how fast it is growing right now, not on average. For that you need the derivative. With polynomials, differentiating is quick and almost mechanical: a single rule is enough. At the end of the page you will come back to this question and answer it.
Power rule and linearity
You already know that the derivative is the limit of the average rate of change as $\Delta x \to 0$. Computing that limit every time would be slow; with polynomials there is a shortcut.
When the polynomial comes as a product, first multiply and then differentiate. For example, $(x+1)(x+2)=x^2+3x+2$, whose derivative is $2x+3$.
Calculate the derivative function of the following polynomial function: $f(x) = 5x^4 - 3x^3 + 7x^2 - 2x + 9$.
Calculate the derivative function of the following polynomial function: $f(x) = 5x^4 - 3x^3 + 7x^2 - 2x + 9$.
Calculate the derivative of the function $p(x) = (2x^2 - 3)(x + 4)$ by first expanding the product of the polynomials.
Calculate the derivative of the function $p(x) = (2x^2 - 3)(x + 4)$ by first expanding the product of the polynomials.
Given the function describing the trajectory of a particle $s(t) = \frac{1}{4}t^4 - \frac{2}{3}t^3 + \frac{1}{2}t^2$, where $s$ is measured in meters and $t$ in seconds,…
Given the function describing the trajectory of a particle $s(t) = \frac{1}{4}t^4 - \frac{2}{3}t^3 + \frac{1}{2}t^2$, where $s$ is measured in meters and $t$ in seconds, determine the expression for its instantaneous velocity $v(t)$.
Value of the derivative at a point
The derivative $f'(x)$ is a function. To find the slope of the tangent line (or the instantaneous rate of change) at a point, substitute its $x$-coordinate into $f'(x)$.
When the problem gives a point on the curve in another way (for example, where it crosses the $y$-axis, that is $x=0$), first find its $x$-coordinate and then evaluate $f'$.
Find the value of the derivative of the function $g(x) = -2x^3 + 4x^2 - x + 5$ at the point with x-coordinate $x = -1$.
Find the value of the derivative of the function $g(x) = -2x^3 + 4x^2 - x + 5$ at the point with x-coordinate $x = -1$.
Determine the gradient of the tangent line to the curve $y = x^3 - 5x + 2$ at the point where it intersects the y-axis.
Determine the gradient of the tangent line to the curve $y = x^3 - 5x + 2$ at the point where it intersects the y-axis.
The profile of a suspension bridge cable is approximated by the parabola $y = 0.01x^2 - 0.6x + 15$, where $x$ and $y$ are measured in meters. Calculate the slope of the…
The profile of a suspension bridge cable is approximated by the parabola $y = 0.01x^2 - 0.6x + 15$, where $x$ and $y$ are measured in meters. Calculate the slope of the cable at the point located $40$ meters from the origin.
Critical points: $f'(x)=0$
You can also ask for the tangent to have a given slope: solve $f'(x)=m$. For example, a tangent parallel to the line $y=x$ has $m=1$; one that makes a $45^\circ$ angle with the positive $x$-axis has $m=\tan 45^\circ=1$.
Determine the stationary points (those where the derivative is zero) of the quadratic function $h(x) = 3x^2 - 12x + 7$.
Determine the stationary points (those where the derivative is zero) of the quadratic function $h(x) = 3x^2 - 12x + 7$.
Find the coordinates of the points on the graph of the function $f(x) = x^3 - 3x^2 + x - 1$ at which the tangent line is parallel to the angle bisector of the first…
Find the coordinates of the points on the graph of the function $f(x) = x^3 - 3x^2 + x - 1$ at which the tangent line is parallel to the angle bisector of the first quadrant.
Find the x-coordinate of the points on the curve $y = \frac{1}{3}x^3 - \frac{5}{2}x^2 + 6x$ where the tangent line forms an angle of $45^\circ$ with the positive x-axis.
Find the x-coordinate of the points on the curve $y = \frac{1}{3}x^3 - \frac{5}{2}x^2 + 6x$ where the tangent line forms an angle of $45^\circ$ with the positive x-axis.
Successive derivatives
Interpretation: $f'$ tells you whether $f$ is increasing or decreasing; $f''$ tells you whether $f'$ is increasing or decreasing, that is, the concavity. If $f''>0$, the graph is concave up; if $f''<0$, concave down. An inflection point is found where $f''$ changes sign (by solving $f''(x)=0$). In physics, if $s(t)$ is the position, then $v(t)=s'(t)$ and $a(t)=s''(t)$.
Calculate the second derivative, $f''(x)$, of the polynomial function $f(x) = x^5 - 4x^4 + 2x^2 - 8$.
Calculate the second derivative, $f''(x)$, of the polynomial function $f(x) = x^5 - 4x^4 + 2x^2 - 8$.
Two test vehicles travel in a straight line. Their positions relative to a reference point are given by $s_1(t) = t^3 - 6t^2 + 15t$ and $s_2(t) = 2t^2 - 4t + 10$, with…
Two test vehicles travel in a straight line. Their positions relative to a reference point are given by $s_1(t) = t^3 - 6t^2 + 15t$ and $s_2(t) = 2t^2 - 4t + 10$, with $t \ge 0$ in seconds and $s$ in meters. Determine at what times both vehicles travel with the same acceleration and calculate the distance separating them at that moment.
Determine the coefficients $a, b, c, d$ of the cubic polynomial function $f(x) = ax^3 + bx^2 + cx + d$ knowing that it passes through the origin, has a stationary point…
Determine the coefficients $a, b, c, d$ of the cubic polynomial function $f(x) = ax^3 + bx^2 + cx + d$ knowing that it passes through the origin, has a stationary point at $x = 2$, and has a point of inflection at $(1, -2)$.
Tangent line and parameters
The tangent to $y=f(x)$ at $x=a$ passes through $(a, f(a))$ and has slope $m=f'(a)$. It is written in point-slope form: $y-f(a)=f'(a)(x-a)$.
When coefficients are missing ($a$, $b$, $c$…), each piece of information gives an equation: "passes through the point $(x_0,y_0)$" gives $f(x_0)=y_0$ and "has slope $m$ at $x_0$" gives $f'(x_0)=m$. With them you form a system.
For a tangent from an external point $P$, call $(t, f(t))$ the point of tangency. The slope is $f'(t)$ and the line must pass through $P$; that gives an equation in $t$. With two curves it works the same way, requiring both to have the same tangent line.
Find the equation of the tangent line to the function $f(x) = -x^2 + 4x - 1$ at its vertex.
Find the equation of the tangent line to the function $f(x) = -x^2 + 4x - 1$ at its vertex.
Determine the values of the parameters $a$ and $b$ such that the parabola $y = ax^2 + bx + 5$ passes through the point $(1, 2)$ and has a tangent line at that point with…
Determine the values of the parameters $a$ and $b$ such that the parabola $y = ax^2 + bx + 5$ passes through the point $(1, 2)$ and has a tangent line at that point with a slope equal to $-1$.
Find the equation of a parabola of the form $y = ax^2 + bx + c$, given that it is tangent to the line $y = -2x + 5$ at the point with x-coordinate $x = 1$ and passes…
Find the equation of a parabola of the form $y = ax^2 + bx + c$, given that it is tangent to the line $y = -2x + 5$ at the point with x-coordinate $x = 1$ and passes through the point $(-1, 15)$.
Find the equations of the tangent lines to the parabola $y = x^2 - 2x + 3$ passing through the exterior point $P(1, -2)$.
Find the equations of the tangent lines to the parabola $y = x^2 - 2x + 3$ passing through the exterior point $P(1, -2)$.
Applications: rate of change and optimization
Differentiating, evaluating, and solving $f'(x)=0$ solve almost all problems: marginal costs, velocities, growth rates, and optimal values.
To optimize a profit, $B(x)=I(x)-C(x)$: differentiate, solve $B'(x)=0$, discard negative solutions, and verify that it is a maximum using the sign of $B'$ (or with $B''<0$).
To find where a function increases or decreases, solve $f'(x)=0$ and study the sign of $f'$ on the resulting intervals. The rate of change is "increasing" or "decreasing" depending on the sign of $f''$; the fastest downward change occurs where $f'$ is at a minimum, that is, where $f''=0$ and $f'$ goes from decreasing to increasing.
To show that a polynomial is always increasing, it is enough to show that $f'(x)>0$ for all $x$. For example, if $f(x)=x^3+2x+5$, then $f'(x)=3x^2+2\geq 2>0$.
Return to the beginning: the speed at which an app is growing today is the derivative $U'(t)$ at that moment. With $U'(5)=63$, that month it was growing at 63 thousand users per month, although more and more slowly.
The production cost, in pesos, of manufacturing $x$ units of a handcrafted product is given by $C(x) = 3\,000 + 240x - 1x^2$. Find the marginal cost function, $C'(x)$.
The production cost, in pesos, of manufacturing $x$ units of a handcrafted product is given by $C(x) = 3\,000 + 240x - 1x^2$. Find the marginal cost function, $C'(x)$.
The number of registered users of a productivity application, in thousands, evolves according to the function $U(t) = -t^3 + 9t^2 + 48t + 200$, where $t$ is the time in…
The number of registered users of a productivity application, in thousands, evolves according to the function $U(t) = -t^3 + 9t^2 + 48t + 200$, where $t$ is the time in months since its launch. Determine the growth rate of the number of users at time $t = 5$ months and state whether at that moment the rate of growth is increasing or decreasing.
A projectile is launched vertically upward. Its height in meters after $t$ seconds is given by $h(t) = -4.9t^2 + 29.4t + 1.5$. Calculate the time it takes to reach its…
A projectile is launched vertically upward. Its height in meters after $t$ seconds is given by $h(t) = -4.9t^2 + 29.4t + 1.5$. Calculate the time it takes to reach its maximum height, knowing that at this point its instantaneous velocity is zero.
The level of noise pollution at an urban junction from 8:00 to 18:00 is modelled by the function $N(t) = 0.1t^3 - 1.5t^2 + 6t + 60$, where $t$ is the time in hours…
The level of noise pollution at an urban junction from 8:00 to 18:00 is modelled by the function $N(t) = 0.1t^3 - 1.5t^2 + 6t + 60$, where $t$ is the time in hours elapsed since 8:00. Determine the time $t$ at which the noise level is decreasing most rapidly.
A company manufactures and sells $x$ units of a tech item. The revenue function is given by $I(x) = 1200x - 2x^2$ and the total cost function is…
A company manufactures and sells $x$ units of a tech item. The revenue function is given by $I(x) = 1200x - 2x^2$ and the total cost function is $C(x) = x^3 - 15x^2 + 400x + 1500$. Determine the number of units that must be produced to maximise profit and calculate this maximum profit.
An environmental study models the concentration of a pollutant in a lake using the function $C(t) = -0.05t^4 + 0.8t^3 - 3.6t^2 + 10$, where $t$ represents the months…
An environmental study models the concentration of a pollutant in a lake using the function $C(t) = -0.05t^4 + 0.8t^3 - 3.6t^2 + 10$, where $t$ represents the months elapsed since the beginning of the year. Find out in which months the concentration of the pollutant is increasing and in which it is decreasing.
Prove mathematically that the polynomial function $f(x) = x^3 + 2x + 5$ is strictly increasing over its entire domain $\mathbb{R}$ and, therefore, has no tangent line…
Prove mathematically that the polynomial function $f(x) = x^3 + 2x + 5$ is strictly increasing over its entire domain $\mathbb{R}$ and, therefore, has no tangent line parallel to the x-axis.
Given the parabolas $f(x) = x^2$ and $g(x) = -x^2 + 8x - 16$, determine the equations of all lines that are tangent to both curves simultaneously.
Given the parabolas $f(x) = x^2$ and $g(x) = -x^2 + 8x - 16$, determine the equations of all lines that are tangent to both curves simultaneously.