What is the derivative of the function 6/(3x^4 + 5) with respect to x?

To find the derivative of the function ( f(x) = \frac{6}{3x^4 + 5} ) with respect to ( x ), we will apply the quotient rule or, more effectively in this case, the chain rule coupled with the power rule.

The function is essentially in the form ( f(x) = 6(3x^4 + 5)^{-1} ). Using the chain rule, we differentiate ( (3x^4 + 5)^{-1} ).

The derivative of ( u^{-1} ) (where ( u = 3x^4 + 5 )) is given by:

[

\frac{d}{dx}(u^{-1}) = -u^{-2} \cdot \frac{du}{dx}

]

Now, let's compute ( \frac{du}{dx} ):

[

\frac{du}{dx} = \frac{d}{dx}(3x^4 + 5) = 12x^3

]

Now we can insert this into our derivative formula:

[

\frac{df}{dx} = 6 \cdot \left(- (3x^4 +