Question

differentiate: f(x) = (x^4+5x^4)^-1

Answer 1

I'll show you how to do it.

Answer 2

This involves the chain rule.

Answer 3

So, if x is inside of some other type of function, you must take the derivatives of what happens to x from inside of the function all the way out. In this case, x becomes the expression (x^4 + 5x^4), and then it enters a function which raises that expression to the power of -1. So we must take the derivative of the expression that x becomes, and multiply it by the derivative of the function that it gets put into (the one that gives it a -1 exponent). This gives the following: \[\frac{d}{dx}(x^4+5x^4)^{-1} = \frac{d}{dx}(x^4+5x^4)\frac{d}{du}(u^{-1})\] where u is the expression (x^4+5x^4)

Answer 4

So this becomes \[(4x^3+20x^3)(-u^{-2})\]

Answer 5

Since u is equal to \[x^4+5x^4\] the final answer is \[(4x^3+20x^3)(-(x^4+5x^4)^{-2})=-\frac{4x^3+20x^3}{(x^4+5x^4)^{2}}\]

Answer 6

Which you could simplify further if you wanted

Answer 7

Thank you very much

Answer 8

No problem

Answer 9

did you finish simplifying it?

Answer 10

\[-\frac{24x^3}{36x^8}=-\frac{2}{3x^5}\]