Question

hw 10 - product and quotient rules section 2.5: problem 5 (1 point) if $f(t)=(t^{2}+3t + 7)(2t^{-2}+5t^{-3})$, find $f(t)$. answer:

Explanation:

Step1: Recall product - rule

The product - rule states that if $y = u\cdot v$, where $u$ and $v$ are functions of $t$, then $y^\prime=u^\prime v + uv^\prime$. Let $u=t^{2}+3t + 7$ and $v = 2t^{-2}+5t^{-3}$.

Step2: Find $u^\prime$ and $v^\prime$

Differentiate $u$ with respect to $t$: $u^\prime=\frac{d}{dt}(t^{2}+3t + 7)=2t + 3$. Differentiate $v$ with respect to $t$: $v^\prime=\frac{d}{dt}(2t^{-2}+5t^{-3})=2\times(-2)t^{-3}+5\times(-3)t^{-4}=-4t^{-3}-15t^{-4}$.

Step3: Apply the product - rule

$f^\prime(t)=u^\prime v+uv^\prime=(2t + 3)(2t^{-2}+5t^{-3})+(t^{2}+3t + 7)(-4t^{-3}-15t^{-4})$.
Expand the first product:

$$
LATEXBLOCK0
$$

Expand the second product:

$$
LATEXBLOCK1
$$

Step4: Combine the two expanded products

$$
LATEXBLOCK2
$$

Answer:

$-11t^{-2}-58t^{-3}-105t^{-4}$