Question

Найдите f'(x), если f(x)=(3-5x) ^{5}

Answer 1

F(x)=(3-5x) ^5
f'(x)=5(3-5x)⁴*(-5)=-25*(3-5x)⁴

Answer 2

To find the derivative of the function $f(x) = (3 - 5x)^5$ with respect to $x$, we can use the chain rule of differentiation. The chain rule states that if we have a function $g(u)$ and another function $f(x) = g(h(x))$, then the derivative of $f(x)$ with respect to $x$ is given by $f'(x) = g'(h(x)) \cdot h'(x)$.

Let's apply the chain rule to $f(x) = (3 - 5x)^5$:

Step 1: Identify the outer function $g(u)$ and the inner function $h(x)$. $g(u) = u^5$, where $u = 3 - 5x$.

Step 2: Find the derivatives of $g(u)$ and $h(x)$. $g'(u) = 5u^{5-1} = 5u^4$. $h'(x) = \frac{d}{dx} (3 - 5x) = -5$ (the derivative of $3$ with respect to $x$ is $0$, and the derivative of $-5x$ is $-5$).

Step 3: Apply the chain rule formula $f'(x) = g'(h(x)) \cdot h'(x)$. $f'(x) = g'(h(x)) \cdot h'(x) = 5(3 - 5x)^4 \cdot (-5)$.

Therefore, the derivative of the function $f(x) = (3 - 5x)^5$ with respect to $x$ is: $f'(x) = -5 \cdot 5(3 - 5x)^4 = -25(3 - 5x)^4.$