Differentiation rules
Polynomials are differentiated by multiplying by the exponent and lowering it by one, but most real functions combine products, quotients and functions nested inside one another: an exponential that depends on time squared, a cost that is a quotient of polynomials. For these combinations there are three fixed rules (product, quotient and chain) which, applied in the right order, differentiate any function without using limits again.
The product rule
Determine the derivative of the function $g(t) = (3t^2 - 5t) \cdot e^t$ using the product rule.
Determine the derivative of the function $g(t) = (3t^2 - 5t) \cdot e^t$ using the product rule.
The quotient rule
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 chain rule
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 $y = \ln(x^2 + 5x + 6)$.
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.
Combining the three rules
Most real functions mix the chain rule with the product rule or the quotient rule. The strategy is always the same: first identify the "outermost" operation (is it a product? a quotient? a single composite function?) and apply its rule; inside each derivative that appears, check again whether the chain rule is needed.
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.
The profile of the roof of an Olympic stadium follows the plane curve given by $f(x) = \frac{100}{x^2 + 25}$, in metres. 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 metres. 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.
Back to the beginning: the exponential that depends on time squared is $f(t) = e^{t^2}$; with the chain rule, its derivative is $f'(t) = e^{t^2} \cdot 2t = 2t\,e^{t^2}$.