question 3 (1 point) find the derivative of the following function. $f(x)=5x^{2}(x + 47)$ $f(x)=15x^{2}-470$…

question 3 (1 point) find the derivative of the following function. $f(x)=5x^{2}(x + 47)$ $f(x)=15x^{2}-470$ $f(x)=15x^{2}+470x$ $f(x)=10x$ $f(x)=5x^{2}+470x$ none of these

question 3 (1 point) find the derivative of the following function. $f(x)=5x^{2}(x + 47)$ $f(x)=15x^{2}-470$ $f(x)=15x^{2}+470x$ $f(x)=10x$ $f(x)=5x^{2}+470x$ none of these

Answer

Explanation:

Step1: Expand the function

First, expand $f(x)=5x^{2}(x + 47)$ using the distributive property $a(b + c)=ab+ac$. So $f(x)=5x^{3}+235x^{2}$.

Step2: Apply the power - rule for derivatives

The power - rule states that if $y = ax^{n}$, then $y^\prime=anx^{n - 1}$. For $y = 5x^{3}$, its derivative $y^\prime_1=5\times3x^{3 - 1}=15x^{2}$. For $y = 235x^{2}$, its derivative $y^\prime_2=235\times2x^{2 - 1}=470x$.

Step3: Find the derivative of the sum

Since $f(x)=5x^{3}+235x^{2}$, then $f^\prime(x)=y^\prime_1 + y^\prime_2=15x^{2}+470x$.

Answer:

$f^{\prime}(x)=15x^{2}+470x$ (the second option)