Question

if (f(x)=(5x + 7)^{-2}), find (f(x)). (-10(5x + 7)^{-3}) find (f(2)). (-.002) question help: video

Explanation:

Step1: Apply chain - rule

Let $u = 5x+7$, then $f(x)=u^{-2}$. The chain - rule states that $f^\prime(x)=\frac{df}{du}\cdot\frac{du}{dx}$. First, find $\frac{df}{du}$: $\frac{df}{du}=-2u^{-3}$ (using the power rule $\frac{d}{du}(u^n)=nu^{n - 1}$ with $n=-2$). And $\frac{du}{dx}=5$.

Step2: Substitute back $u$

$f^\prime(x)=-2(5x + 7)^{-3}\cdot5=-10(5x + 7)^{-3}$

Step3: Evaluate $f^\prime(2)$

Substitute $x = 2$ into $f^\prime(x)$: $f^\prime(2)=-10(5\times2 + 7)^{-3}=-10(10 + 7)^{-3}=-10\times17^{-3}=-10\times\frac{1}{4913}\approx - 0.002035\approx - 0.002$

Answer:

$f^\prime(x)=-10(5x + 7)^{-3}$; $f^\prime(2)\approx - 0.002$