Differentiation Rules
A video goes viral and its views grow like $e^{t^2}$, where $t$ is the time in days. How fast are the views growing at each instant? That function is not a polynomial: it has an exponential with another function inside it. With limits it would take a long time, but with three rules (product, quotient, and chain) you can differentiate almost any combination.
Basic derivatives
Before combining functions you need the derivatives of the individual pieces. The derivative of $x^n$ is obtained with the four-step rule; let's look at one case to remember how it works.
To differentiate powers with fractions or roots, first write them as exponents: $\dfrac{1}{x^3}=x^{-3}$ and $\sqrt{x}=x^{1/2}$.
| $f(x)$ | $f'(x)$ |
|---|---|
| $e^x$ | $e^x$ |
| $a^x$ | $a^x \ln a$ |
| $\ln x$ | $\dfrac{1}{x}$ |
| $\operatorname{sen} x$ | $\cos x$ |
| $\cos x$ | $-\operatorname{sen} x$ |
| $\operatorname{arcsen} x$ | $\dfrac{1}{\sqrt{1-x^2}}$ |
| $\arctan x$ | $\dfrac{1}{1+x^2}$ |
Calculate the derivative function of $f(x) = 5x^4 - \frac{3}{x^2} + 2\sqrt{x} - 7$, applying the basic power rules of differentiation and linearity.
Calculate the derivative function of $f(x) = 5x^4 - \frac{3}{x^2} + 2\sqrt{x} - 7$, applying the basic power rules of differentiation and linearity.
Calculate the derivative of the function $f(x) = 3^{2x} + \arctan(x)$, recalling the differentiation rule for exponentials with a base other than $e$ and for inverse…
Calculate the derivative of the function $f(x) = 3^{2x} + \arctan(x)$, recalling the differentiation rule for exponentials with a base other than $e$ and for inverse trigonometric functions.
Product rule
Is the derivative of a product the product of the derivatives? Let's test it with $x^2\cdot x^3=x^5$, whose derivative is $5x^4$. If we multiplied the derivatives we would get $2x\cdot3x^2=6x^3$, which doesn't match. It doesn't work with $x\cdot x=x^2$ either: it would give $1\cdot1=1$ and the actual derivative is $2x$.
Notice what does work: $2x\cdot x^3+x^2\cdot3x^2=2x^4+3x^4=5x^4$. Each factor is differentiated in turn while the other is left unchanged.
Determine the derivative of the function $g(t) = (3t^2 - 5t) \times e^t$ using the product rule.
Determine the derivative of the function $g(t) = (3t^2 - 5t) \times e^t$ using the product rule.
The number of daily streams of a viral song evolves according to the function $R(t) = 50 t^2 e^{-0.2t}$, where $t$ is measured in days since its release and $R(t)$ in…
The number of daily streams of a viral song evolves according to the function $R(t) = 50 t^2 e^{-0.2t}$, where $t$ is measured in days since its release and $R(t)$ in thousands of streams. Calculate the instantaneous rate of change $R'(t)$ by applying differentiation rules and state its fully factorised expression.
Quotient rule
Let's discover the rule with a case we already know how to solve. Take $\dfrac{x}{x^2}=x^{-1}$, whose derivative is $-x^{-2}=-\dfrac{1}{x^2}$.
If we thought the derivative of a quotient was the quotient of the derivatives, we would get $\dfrac{1}{2x}$, which doesn't match. So let's try the product pattern, writing $\dfrac{f}{g}=f\cdot g^{-1}$ with $f=x$ and $g=x^2$.
With the product rule: $f'g^{-1}+f\,(g^{-1})'=\dfrac{1}{x^2}+x\cdot\left(-\dfrac{2x}{x^4}\right)=\dfrac{1}{x^2}-\dfrac{2}{x^2}=-\dfrac{1}{x^2}$. This one does match.
Rearranging, that sum is $\dfrac{f'g-fg'}{g^2}$. With our data: $\dfrac{1\cdot x^2-x\cdot2x}{x^4}=\dfrac{-x^2}{x^4}=-\dfrac{1}{x^2}$.
An application: the tangent line
The derivative at a point is the slope of the tangent line. With the quotient rule you can find the tangent to a rational curve.
Apply the quotient rule to find the derivative of the rational function $h(x) = \frac{2x + 1}{x^2 - 3}$.
Apply the quotient rule to find the derivative of the rational function $h(x) = \frac{2x + 1}{x^2 - 3}$.
The average production cost of a high-precision electronic component is given by $C(x) = \frac{\ln(x + 1)}{x}$, where $x$ is the number of batches manufactured (…
The average production cost of a high-precision electronic component is given by $C(x) = \frac{\ln(x + 1)}{x}$, where $x$ is the number of batches manufactured ($x > 0$). Determine the function that describes the rate of change of the average cost, $C'(x)$.
The profile of the roof of an Olympic stadium follows the plane curve given by $f(x) = \frac{100}{x^2 + 25}$, in meters. Calculate the general equation of the tangent…
The profile of the roof of an Olympic stadium follows the plane curve given by $f(x) = \frac{100}{x^2 + 25}$, in meters. Calculate the general equation of the tangent line at the point with x-coordinate $x = 5$ to design a straight structural support that rests tangentially at that point.
Chain rule
A composite function has "layers": $\operatorname{sen}(4x^3-x)$ is a sine (outer layer) applied to $4x^3-x$ (inner layer).
When $u$ is a fraction, as in $\operatorname{arcsen}\!\left(\frac{x-1}{x+1}\right)$ with $x>1$, you first compute $u'$ with the quotient rule and then simplify $1-u^2$ with algebra. Watch the domain: the arcsine only exists if $-1\le u\le1$, and for $x>1$ the fraction $\frac{x-1}{x+1}$ is between $0$ and $1$, so it can be differentiated. To simplify, your notes on algebraic fractions will help.
Calculate the derivative of the composite function $f(x) = \sin(4x^3 - x)$, applying the chain rule correctly.
Calculate the derivative of the composite function $f(x) = \sin(4x^3 - x)$, applying the chain rule correctly.
Find the first derivative of the logarithmic function $y = \ln(x^2 + 5x + 6)$.
Find the first derivative of the logarithmic function $y = \ln(x^2 + 5x + 6)$.
An articulated robotic arm moves in such a way that the angle of its main joint (in radians), as a function of a control parameter $x$, is…
An articulated robotic arm moves in such a way that the angle of its main joint (in radians), as a function of a control parameter $x$, is $\theta(x) = \arcsin\left(\frac{x-1}{x+1}\right)$ for $x > 1$. Obtain and state the fully simplified form of $\theta'(x)$.
Combining the rules
Most functions mix several rules. The strategy is always the same: identify the outermost operation (product? quotient? composite function?), apply its rule, and inside each derivative that appears, ask yourself again whether the chain rule is needed.
Exponential times sine: checking an equation
A product like $e^{-\alpha t}\operatorname{sen}(\omega t)$ appears in circuits and vibrations. It is differentiated with the product and chain rules, and the constants $\alpha$ and $\omega$ come out as multipliers from the chain rule.
With data from a graph
Sometimes you don't have the complete formula, only values of $f$, $g$, and their derivatives at certain points, or their graphs. If the graphs are a parabola and a line, you can read off the points and compute the derivatives with the rules above. The rules still work.
The temperature of an industrial oven after being switched off drops according to the cooling law $T(t) = 20 + 80 \cos^2\left(\frac{\pi t}{4}\right)$, with $T$ in…
The temperature of an industrial oven after being switched off drops according to the cooling law $T(t) = 20 + 80 \cos^2\left(\frac{\pi t}{4}\right)$, with $T$ in degrees Celsius and $t$ in hours. Find the expression for the cooling rate $T'(t)$ using the chain rule.
Let $f(x)$ and $g(x)$ be two functions whose graphs are shown in the figure. The graph of $f$ is the parabola $f(x)=-x^2+6x-3$, which passes through the points $(2,\,5)$…
Let $f(x)$ and $g(x)$ be two functions whose graphs are shown in the figure. The graph of $f$ is the parabola $f(x)=-x^2+6x-3$, which passes through the points $(2,\,5)$ and $(4,\,5)$. The graph of $g$ is the line that passes through the points $(2,\,4)$ and $(4,\,2)$. The composite function $h(x)=f(g(x))$ and the product function $p(x)=f(x)\times g(x)$ are defined.
Calculate the exact values of:
- a) $h'(2)$
- b) $p'(4)$
In an AC RLC circuit, the charge on the capacitor is given by the function $q(t) = e^{-\alpha t} (A \sin(\omega t) + B \cos(\omega t))$. Show, by calculating $q'(t)$ and…
In an AC RLC circuit, the charge on the capacitor is given by the function $q(t) = e^{-\alpha t} (A \sin(\omega t) + B \cos(\omega t))$. Show, by calculating $q'(t)$ and $q''(t)$ in detail, that for the constants $A=1$, $B=0$, $\alpha=2$, and $\omega=3$, the function satisfies the second-order differential equation:
$$ q''(t) + 4q'(t) + 13q(t) = 0 $$Simplify before differentiating
With logarithms, the properties $\ln(ab)=\ln a+\ln b$, $\ln\frac{a}{b}=\ln a-\ln b$, and $\ln a^n=n\ln a$ can turn a hard derivative into an easy one.
If trigonometric functions appear inside the logarithm, at the end you will need identities to simplify the result.
The sound pressure level in an anechoic chamber, after applying a frequency filter, is given by the function: $$ L(x) = \ln\left( \sqrt{\frac{1 + \sin(2x)}{1 -…
The sound pressure level in an anechoic chamber, after applying a frequency filter, is given by the function:
$$ L(x) = \ln\left( \sqrt{\frac{1 + \sin(2x)}{1 - \sin(2x)}} \right) $$Show, using the laws of logarithms before differentiating and applying trigonometric identities in the final result, that $L'(x) = \frac{2}{\cos(2x)}$.
Implicit differentiation
Sometimes the curve comes as an equation in $x$ and $y$ that you can't easily solve for $y$, like $x^2+y^2=25$. The idea is to differentiate both sides with respect to $x$, remembering that $y$ depends on $x$. So $y^2$ is a composite function: $(y^2)'=2y\cdot y'$.
The trajectory of a satellite during a complex orbital maneuver is described in a reference plane by the curve implicitly defined by the equation $x^2 y - 3y^3 x = 10$.…
The trajectory of a satellite during a complex orbital maneuver is described in a reference plane by the curve implicitly defined by the equation $x^2 y - 3y^3 x = 10$. Calculate the value of the slope of the trajectory, $y'$, at the point with coordinates $(-2, 1)$.
Successive derivatives
If $f'$ is a function, it can also be differentiated. This gives the second derivative $f''$, then $f'''$, and in general $f^{(n)}$. The second derivative measures how the rate of change changes: if $s$ is position, $s'$ is velocity and $s''$ is acceleration.
Determine the second derivative, $f''(t)$, of the function $f(t) = \cos(5t) - t^3$.
Determine the second derivative, $f''(t)$, of the function $f(t) = \cos(5t) - t^3$.
The position of a pneumatic piston inside a cylinder is described by the equation of motion $s(t) = \ln(1 + t^2)$, where $s$ is in decimeters and $t$ in seconds.…
The position of a pneumatic piston inside a cylinder is described by the equation of motion $s(t) = \ln(1 + t^2)$, where $s$ is in decimeters and $t$ in seconds. Calculate the expression for the acceleration of the piston, corresponding to the second derivative $s''(t)$, and determine at what time $t > 0$ this acceleration becomes zero.
The intensity of an electromagnetic signal attenuated as it passes through a dense medium follows the function $I(x) = x e^{-x}$. Calculate the first three derivatives…
The intensity of an electromagnetic signal attenuated as it passes through a dense medium follows the function $I(x) = x e^{-x}$. Calculate the first three derivatives $I'(x)$, $I''(x)$, and $I'''(x)$, and from the results obtained, deduce and write down the general formula for the nth derivative $I^{(n)}(x)$.
Back to the beginning: the views $e^{t^2}$ have an outer layer ($e^u$) and an inner layer ($u=t^2$). With the chain rule, the growth rate is $2t\,e^{t^2}$. It grows faster and faster, because both $t$ and $e^{t^2}$ increase over time.