Question

Найти производную f(x)=(3x^4-5)^6

Answer 1

Решение приложено к снимку:

Answer 2

To find the derivative of the function $f(x) = (3x^4 - 5)^6$ with respect to $x$, we can use the chain rule. The chain rule states that if we have a composite function $u(v(x))$, then its derivative is given by $u'(v(x)) \cdot v'(x)$.

Let's apply the chain rule step by step:

Step 1: Identify the inner function $u$ and the outer function $v$. Let $v(x) = 3x^4 - 5$, and $u(v) = v^6$.

Step 2: Find the derivative of the inner function $v(x)$. The derivative of $v(x) = 3x^4 - 5$ with respect to $x$ is: $v'(x) = \frac{d}{dx} (3x^4 - 5) = 12x^3.$

Step 3: Find the derivative of the outer function $u(v)$. The derivative of $u(v) = v^6$ with respect to $v$ is: $u'(v) = \frac{d}{dv} (v^6) = 6v^5.$

Step 4: Apply the chain rule by multiplying the derivatives found in Steps 2 and 3. The derivative of $f(x) = (3x^4 - 5)^6$ with respect to $x$ is: $f'(x) = u'(v(x)) \cdot v'(x) = 6(3x^4 - 5)^5 \cdot 12x^3.$

Now we have the derivative of the function $f(x)$: $f'(x) = 72x^3(3x^4 - 5)^5.$