To find the derivative of the product of the two functions [tex]\( (10 x^5 + 7 x^9)(3 e^x - 5) \)[/tex] using the product rule, we follow these steps.
First, let's recall the product rule for differentiation. If we have two functions [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex], the derivative of their product is given by:
[tex]\[
(f \cdot g)' = f' \cdot g + f \cdot g'
\][/tex]
Here, let's identify our functions [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex]:
[tex]\[
f(x) = 10 x^5 + 7 x^9
\][/tex]
[tex]\[
g(x) = 3 e^x - 5
\][/tex]
Next, we need to compute the derivatives [tex]\( f'(x) \)[/tex] and [tex]\( g'(x) \)[/tex].
1. Compute [tex]\( f'(x) \)[/tex]:
[tex]\[
f(x) = 10 x^5 + 7 x^9
\][/tex]
[tex]\[
f'(x) = \frac{d}{dx}(10 x^5) + \frac{d}{dx}(7 x^9)
\][/tex]
Using the power rule [tex]\(\frac{d}{dx}(x^n) = n x^{n-1}\)[/tex], we get:
[tex]\[
f'(x) = 10 \cdot 5 x^{4} + 7 \cdot 9 x^{8}
\][/tex]
[tex]\[
f'(x) = 50 x^{4} + 63 x^{8}
\][/tex]
2. Compute [tex]\( g'(x) \)[/tex]:
[tex]\[
g(x) = 3 e^x - 5
\][/tex]
The derivative of [tex]\( e^x \)[/tex] is [tex]\( e^x \)[/tex]. Thus, we get:
[tex]\[
g'(x) = 3 \frac{d}{dx}(e^x) - \frac{d}{dx}(5)
\][/tex]
[tex]\[
g'(x) = 3 e^x
\][/tex]
Now, apply the product rule:
[tex]\[
(f \cdot g)' = f' \cdot g + f \cdot g'
\][/tex]
Substitute [tex]\( f(x) \)[/tex], [tex]\( f'(x) \)[/tex], [tex]\( g(x) \)[/tex], and [tex]\( g'(x) \)[/tex] into the formula:
[tex]\[
\left((10 x^5 + 7 x^9)(3 e^x - 5)\right)' = (50 x^4 + 63 x^8)(3 e^x - 5) + (10 x^5 + 7 x^9)(3 e^x)
\][/tex]
Thus, the derivative of the given function is:
[tex]\[
(63 x^8 + 50 x^4)(3 e^x - 5) + 3 (7 x^9 + 10 x^5) e^x
\][/tex]
This is the unexpanded form of the derivative using the product rule.
