Ok, let's work on finding the derivative of \(f(x)=2(3x^7 -7x^6 )^{13}\).
To compute this derivative, let's use the chain rule which states that the derivative of a composition of functions is the product of the derivative of the outer function and the derivative of the inner function.
Let's define \(u = 2(3x^7 -7x^6 )\) and \(v = u^{13}\).
First, let's differentiate the outer function \(v\), treating \(u\) as an independent variable. The derivative of \(v\) is \\(v' = 13u^{12}\).
Now, let's find the derivative of the inner function \(u\). To do that, we differentiate the function \(u = 2(3x^7 -7x^6 )\) with respect to \(x\), yielding \\(u' = 2(21x^6 - 42x^5)\), which simplifies to \(u' = 42x^6 - 84x^5\).
Now that we have both derivatives, we can use the chain rule to find the derivative of the composite function. According to the chain rule, the derivative of the composition of \(u\) and \(v\) is given by \(v' * u'\).
By calculating the product of these derivatives, we find the derivative of the given function: \(2*(273x^6 - 546x^5)*(3x^7 - 7x^6)^{12}\).
So, the first derivative of the function \(f(x)=2(3x^7 -7x^6 )^{13}\) = \(2*(273x^6 - 546x^5)*(3x^7 - 7x^6)^{12}\).