Question

Calculate the derivative of the function.

t(x) = [(x + 1)^0.6-9x]^-5.1

t'(x) = ?

Answer 1

This is the function:\[t(x)=((x+1)^{0.6} -9x)^{-5.1}\]
We can see several steps in this function, so we need the Chain Rule.
Basically, the CR says: differentiate the last step first, then multiply with the derivative of the step before etc.
Because the steps we see here are all power functions, the rule to differentiate them is:\[(x^n)' =nx^{n-1}\]OK, let's do it:\[t'(x)=-5.1((x+1)^{0.6} -9x)^{-6.1}*(0.6(x+1)^{-0.4} -9)*1\]We could try to tidy this formula up a bit, but I wouldn't bother: the original function was a mess to begin with...;)

Answer 2

yea i can see that. so i would have to calculate that to in order to find the answer?

Answer 3

Is this a 'chain rule' problem?
A trick to nested functions is to make a new function out of every expression in parenthesis.

Answer 4

yes it is, im just really confused on how to get it problem going

Answer 5

\[
\begin{array}{rcl}
t(x) &=& [(x + 1)^{0.6}-9x]^{-5.1} \\
u(x) &=& (x + 1)^{0.6}-9x \\
v(x) &=& x+1
\end{array}
\]So consider it like this first.

Answer 6

First find derivative of \(v(x)\) then \(u(x)\) and finally your main goal.

Answer 7

The answer is the function in my reply above.
I don't think you are supposed to rewrite t(x) in terms of other functions before differentiating, because the function is so "badly" designed (the exponents 0.6 and -5.1 are not promising...)
@wio has done a nice job of visualising the steps of this function.
I myself always try to differentiate such a function directly, without using u and v and so on. You have to decide yourself what to do, in the end the result is the same!

Answer 8

ok thanks a lot for the help guys i will look this over :)