Question

find the derivative of the function. f(t) = 4(t^7 - 6)^5 f(t) =

Explanation:

Step1: Apply constant - multiple rule

The constant - multiple rule states that if $y = cf(x)$, then $y'=cf'(x)$. Here $c = 4$ and $y=(t^{7}-6)^{5}$, so $f'(t)=4\frac{d}{dt}(t^{7}-6)^{5}$.

Step2: Apply chain - rule

Let $u=t^{7}-6$, then $y = u^{5}$. The chain - rule $\frac{dy}{dt}=\frac{dy}{du}\cdot\frac{du}{dt}$. First, find $\frac{dy}{du}$: $\frac{dy}{du}=\frac{d}{du}(u^{5}) = 5u^{4}$. Second, find $\frac{du}{dt}$: $\frac{du}{dt}=\frac{d}{dt}(t^{7}-6)=7t^{6}$.

Step3: Substitute $u$ back and calculate

Since $\frac{dy}{dt}=\frac{dy}{du}\cdot\frac{du}{dt}$, substituting $u = t^{7}-6$ back, we have $\frac{d}{dt}(t^{7}-6)^{5}=5(t^{7}-6)^{4}\cdot7t^{6}=35t^{6}(t^{7}-6)^{4}$. Then $f'(t)=4\times35t^{6}(t^{7}-6)^{4}=140t^{6}(t^{7}-6)^{4}$.

Answer:

$140t^{6}(t^{7}-6)^{4}$