Question

Найти f’(x) если f(x)=(5-4/x)^2

Answer 1

На сколько я помню, получается ((-32)/x)-40

Answer 2

f’(x)=2(5-4/х)*4/х^2=(10-8/x)*4/x^2

Answer 3

To find the derivative of the function $f(x) = \left(5 - \frac{4}{x}\right)^2$, we'll use the chain rule and the power rule of differentiation. The chain rule states that if we have a function of the form $u(x)^n$, then its derivative is $n \cdot u(x)^{n-1} \cdot u'(x)$, where $u'(x)$ is the derivative of $u(x)$ with respect to $x$.

Let's go step-by-step to find $f'(x)$:

Step 1: Identify the inner function $u(x)$ and the outer function $v(x)$. $u(x) = 5 - \frac{4}{x}$ $v(x) = u(x)^2$

Step 2: Find the derivative of $u(x)$ with respect to $x$. $u'(x) = \frac{d}{dx}\left(5 - \frac{4}{x}\right) = 0 + \frac{4}{x^2} = \frac{4}{x^2}$

Step 3: Apply the chain rule and the power rule. $f'(x) = v'(x) = 2 \cdot u(x)^{2-1} \cdot u'(x) = 2 \cdot \left(5 - \frac{4}{x}\right)^{2-1} \cdot \frac{4}{x^2}$

Simplify the expression: $f'(x) = 2 \cdot \left(5 - \frac{4}{x}\right) \cdot \frac{4}{x^2}$

Now, we can leave it as is, or if you want, we can further simplify it:

$f'(x) = \frac{8}{x^2} \cdot \left(5x - 4\right)$

So, the derivative of the function $f(x) = \left(5 - \frac{4}{x}\right)^2$ is $f'(x) = \frac{8}{x^2} \cdot \left(5x - 4\right)$.