Question

Evaluate the difference quotient for the given function.

Answer 1

Assuming you mean f(t) = g(t) × h(t), notice that

f(t) = g(t) × h(t) = cos(t) sin(t) = 1/2 sin(2t)

Then the difference quotient of f is

$\dfrac{\frac12 \sin(2(t+h)) - \frac12 \sin(2t)}h = \dfrac{\sin(2t+2h) - \sin(2t)}{2h}$

Recall the angle sum identity for sine:

sin(x + y) = sin(x) cos(y) + cos(x) sin(y)

Then we can write the difference quotient as

$\dfrac{\sin(2t)\cos(2h) + \cos(2t)\sin(2h) - \sin(2t)}{2h}$

or

$\boxed{\sin(2t)\dfrac{\cos(2h)-1}{2h} + \cos(2t)\dfrac{\sin(2h)}{2h}}$

(As a bonus, notice that as h approaches 0, we have (cos(2h) - 1)/(2h) → 0 and sin(2h)/(2h) → 1, so we recover the derivative of f(t) as cos(2t).)