Chain and product rules

Functions built from functions

Many physics formulas are not simple powers or sums. They may involve one function inside another, such as

$sin(omega t)$

or

$e^{-t/RC}$

These are composite functions. The chain rule tells us how to differentiate them.

The chain rule

If $y=f(u)$ and $u=g(x)$, then $y=f(g(x))$. The derivative is

rac{dy}{dx} = rac{dy}{du}rac{du}{dx}

In prime notation:

rac{d}{dx}f(g(x)) = f'(g(x))g'(x)

The idea is that the outside function changes with respect to the inside function, and the inside function changes with respect to the original variable.

Chain rule example

Suppose

$x(t)=sin(3t)$

The outside function is sine, and the inside function is $3t$. Differentiate the outside:

$cos(3t)$

Then multiply by the derivative of the inside, which is 3:

rac{dx}{dt}=3cos(3t)

The factor 3 matters. It represents how quickly the input angle changes with time.

Chain rule in physics

The chain rule appears constantly in oscillations, waves, decay, and circular motion. If

$q(t)=Q_0e^{-t/RC}$

then the derivative is

rac{dq}{dt}=-rac{1}{RC}Q_0e^{-t/RC}

The factor $-1/(RC)$ comes from differentiating the exponent. It controls the decay rate.

Products of changing quantities

Sometimes two changing quantities are multiplied. For example, momentum is

$p = mv$

If mass is constant, $dp/dt = m,dv/dt$. But in systems such as rockets, mass may change with time, so both factors matter.

The product rule says

rac{d}{dx}[f(x)g(x)] = f'(x)g(x)+f(x)g'(x)

Both terms are needed because either factor may be changing.

Product rule example

Let

$h(t)=t^2sin t$

Then

rac{dh}{dt}=2tsin t + t^2cos t

The first term comes from differentiating $t^2$ while leaving $sin t$ alone. The second comes from leaving $t^2$ alone while differentiating $sin t$.

Quotients

The quotient rule is useful for ratios:

ight)=rac{g f' - f g'}{g^2}$$ However, many quotient problems can also be handled by rewriting with negative powers. For example, $$rac{1}{x^2}=x^{-2}$$ then use the power rule. ## Common physics pattern: sinusoidal motion For simple harmonic motion, $$x(t)=Acos(omega t + phi)$$ Using the chain rule, $$v(t)=rac{dx}{dt}=-Aomegasin(omega t + phi)$$ Differentiating again, $$a(t)=-Aomega^2cos(omega t + phi)$$ Since $x(t)=Acos(omega t + phi)$, $$a(t)=-omega^2x(t)$$ This relationship is central to oscillations. ## The big idea The chain rule handles functions inside functions. The product rule handles products of changing quantities. These rules are essential because real physics functions often combine motion, time dependence, geometry, exponentials, and trig functions. Mastering them turns calculus into a practical tool for modeling nature.