Question

У=(X-6)^(X-3)+5 ПРОИЗВОДНАЯ

Answer 1

Дана функция:

$y = (x - 6)^{x - 3} + 5$

Запишем функцию в другом виде:

$y = {e}^{ ln((x - 6)^{x - 3} ) } + 5 \\ y = {e}^{(x - 3) ln(x - 6) } + 5$

Берём производную:

$y' = ( {e}^{(x - 3) ln(x - 6) } + 5)' = ( {e}^{(x - 3) ln(x - 6) } )' = \\ = ( {e}^{f} )' \times( (x - 3) ln(x - 6) )' = {e}^{f} ( ln(x - 6) + \frac{x - 3}{x - 6} ) = \\ = {e}^{f} \frac{(x - 6) ln(x - 6) + x - 3 }{x - 6} = \\ = \frac{(x - 6)^{x - 3} (x - 3 + (x - 6) ln(x - 6)) }{x - 6}$

Answer 2

To find the derivative of the function $У=(X-6)^{X-3}+5$ with respect to $X$, we can use the chain rule and the power rule. The chain rule states that if we have a composite function $f(g(X))$, then its derivative is given by $f'(g(X)) \cdot g'(X)$. The power rule states that the derivative of $x^n$ with respect to $x$ is $n \cdot x^{(n-1)}$. Now, let's find the derivative:

Step 1: Find the derivative of the first term $(X-6)^{X-3}$ Let $u = X-6$ and $v = X-3$. Now the function becomes $У = u^v + 5$.

Using the chain rule, the derivative of the first term is: $\frac{d}{dX}(u^v) = v \cdot u^{(v-1)} \cdot \frac{du}{dX}$.

Step 2: Find $\frac{du}{dX}$ $u = X - 6$, so $\frac{du}{dX} = 1$.

Step 3: Find $\frac{d}{dX}(X-6)^{X-3}$ Using the power rule and the chain rule together, we get: $\frac{d}{dX}(X-6)^{X-3} = (X-3) \cdot (X-6)^{(X-3)-1} \cdot 1 = (X-3) \cdot (X-6)^{(X-4)}$.

Step 4: Find the derivative of the constant term, which is simply 0.

Now, we can write the overall derivative: $\frac{dУ}{dX} = (X-3) \cdot (X-6)^{(X-4)} + 0$.

So, the derivative of the function $У=(X-6)^{X-3}+5$ with respect to $X$ is $(X-3) \cdot (X-6)^{(X-4)}$.