Question

Найти производную сложной функции: y=5/(10-5x)^6

Answer 1

Y'=0×(10-5x)^6-5×((10-5x)^6)'/((10-5x)^8=25×(10-5x)^5/(10-5x)^8=25/(10-5x)^3

Answer 2

To find the derivative of the given composite function $y = \frac{5}{{(10 - 5x)}^6}$, we can use the chain rule. The chain rule states that for a composite function $f(g(x))$, the derivative is given by $\frac{{df}}{{dx}} = \frac{{df}}{{dg}} \cdot \frac{{dg}}{{dx}}$.

Let's break down the function into two parts:

1. The outer function: $f(u) = \frac{5}{{u^6}}$, where $u = 10 - 5x$.
2. The inner function: $u = g(x) = 10 - 5x$.

Now, let's find the derivatives step by step:

1. Derivative of the outer function $f(u)$: $\frac{{df}}{{du}} = -6 \cdot \frac{5}{{u^7}} = -\frac{30}{{u^7}}$

2. Derivative of the inner function $u = g(x)$: $\frac{{du}}{{dx}} = -5$

Now, we can apply the chain rule:

$\frac{{dy}}{{dx}} = \frac{{df}}{{du}} \cdot \frac{{du}}{{dx}}$ $\frac{{dy}}{{dx}} = -\frac{30}{{(10 - 5x)^7}} \cdot (-5)$ $\frac{{dy}}{{dx}} = \frac{{150}}{{(10 - 5x)^7}}$

So, the derivative of the given composite function is: $\frac{{dy}}{{dx}} = \frac{{150}}{{(10 - 5x)^7}}$