Question

suppose y=(x^3 + 7x)(4x^2 + 9). here are two different ways to compute the derivative of this function.
a. first, expand the given product and then compute the derivative.
y=
y=
b. first, apply the product rule and then simplify the result.
y=fg + fg=(
)cdot(4x^2 + 9)+(
)cdot(8x)
y=
(simplified)

Explanation:

Step1: Expand the product

$$
LATEXBLOCK0
$$

Step2: Differentiate the expanded function

Using the power - rule $\frac{d}{dx}(x^{n})=nx^{n - 1}$, we have:

$$
LATEXBLOCK1
$$

Step3: Apply the product rule

Let $f(x)=x^{3}+7x$ and $g(x)=4x^{2}+9$. Then $f'(x)=3x^{2}+7$ and $g'(x)=8x$.
By the product rule $y'=f'g + fg'=(3x^{2}+7)(4x^{2}+9)+(x^{3}+7x)(8x)$.

Step4: Expand and simplify the product - rule result

$$
LATEXBLOCK2
$$

Answer:

a. $y = 4x^{5}+37x^{3}+63x$, $y'=20x^{4}+111x^{2}+63$
b. $y'=(3x^{2}+7)(4x^{2}+9)+(x^{3}+7x)(8x)$, $y'=20x^{4}+111x^{2}+63$