Calculus - Derivative of many exponents

[jazxd]

1. Homework Statement
Find the derivative.

2. Homework Equations
h(t)= t^(pi^3)+(pi^3)^t+(pi)t^33. The Attempt at a Solution
- i know that pi is a constant (number) so when finding it's derivative, it will be something like, (2nd part= (pi^3)^t ) : ln pi (pi^3)^t * 3t
- 3rd part = (pi)t^3 : 3(pi)t^2

- But the part I'm most confused is the first part = t^(pi^3)
- Would it be : 3pi(t^pi^3)*3pi ?

p.s: first part doesn't have any parenthesis around in the equation ( t^pi^3)
and sorry for such a messy equations. This is my first time using this forum.

ANSWER: pi^3)t^(pi^3-1) + ln (pi^3)^t + 3(pi)t^2
but i don't know how to.. get that

 

[tiny-tim]

welcome to pf!

hi jazxd! welcome to pf!

(have a pi: π and try using the X² icon just above the Reply box )

- i know that pi is a constant (number) so when finding it's derivative, it will be something like, (2nd part= (pi^3)^t ) : ln pi (pi^3)^t * 3t

not quite …

hint: rewrite (π³)^t as (e^{ln (π3)})^t = e^{t(ln (π3))}

alternatively as π^3t = (e^{ln π})^3t = e^{3t(ln π)}

- But the part I'm most confused is the first part = t^(pi^3)
- Would it be : 3pi(t^pi^3)*3pi ?

no … this is just tⁿ with n = π³

 

[jazxd]

hmm, i know n is a number so it would require ln but why also an e^ln ?
and doesn't e and ln cancel out @__@;
im sorry, I am only a senior in high school, and these stuff gets me big time.

so can you briefly show me how to do these 3 parts ?
im more like, "if i see the answer (or the works) first, i get it afterward" type learner

 

[tiny-tim]

hi jazxd!

(just got up :zzz: …)

hmm, i know n is a number so it would require ln but why also an e^ln ?

because you know how to differentiate e^t, but you don't know how to differentiate n^t

(or, like me, even if you do know, you can't be sure you remember it right! )

and doesn't e and ln cancel out @__@;

yup! … that's why it works!

now try it yourself ​