question 4 (1 point) find the derivative of the following function. $f(x)=\frac{5x^{2}}{x + 47}$ $f(x)=10x$…

question 4 (1 point) find the derivative of the following function. $f(x)=\frac{5x^{2}}{x + 47}$ $f(x)=10x$ $f(x)=\frac{5x^{2}+470x}{(x + 47)^{2}}$ $f(x)=\frac{5x^{2}-470x}{(x + 47)^{2}}$ none of these

question 4 (1 point) find the derivative of the following function. $f(x)=\frac{5x^{2}}{x + 47}$ $f(x)=10x$ $f(x)=\frac{5x^{2}+470x}{(x + 47)^{2}}$ $f(x)=\frac{5x^{2}-470x}{(x + 47)^{2}}$ none of these

Answer

Explanation:

Step1: Apply quotient - rule

The quotient - rule states that if $y=\frac{u}{v}$, then $y^\prime=\frac{u^\prime v - uv^\prime}{v^{2}}$. Here, $u = 5x^{2}$, so $u^\prime=10x$, and $v=x + 47$, so $v^\prime = 1$.

Step2: Substitute into quotient - rule formula

$f^\prime(x)=\frac{(10x)(x + 47)-5x^{2}(1)}{(x + 47)^{2}}$.

Step3: Expand the numerator

$(10x)(x + 47)-5x^{2}=10x^{2}+470x-5x^{2}=5x^{2}+470x$. So, $f^\prime(x)=\frac{5x^{2}+470x}{(x + 47)^{2}}$.

Answer:

$f(x)=\frac{5x^{2}+470x}{(x + 47)^{2}}$