Calculating the Derivative of w with Respect to t

[oswald]

Homework Statement

Finda dw/dt
w=3xy/x²-y²
x=t³
y=e^2t

Homework Equations

w=(3t³e^2t)/(t⁶-e^4t)

The Attempt at a Solution

Well,is there anothe way to solve this, instead of dw/dt; like dw/dx * dw/dt + dw/dy * dy/dt ?

 

[Dick]

Sure. You could just the chain rule as well. You should get the same answer both ways. Just substituting like you did looks to be a little easier.

 

[oswald]

Chain Rule Exponential and logarithmic

f(x) = ln [ e^ln(x+1) ]
f' = ?

 

[Mark44]

Simplify it a bit first to make your differentiation easier. What is ln(e^whatever)?

 

[oswald]

f(x) = ln { e^[ln(x+1)] }

well, i have this answer, but i don't understand
ln [ e^ln(x+1) ] = ln(x+1)

f'(x) = 1/(e^ln(x+1)) * e^ln(x+1) * 1/(x+1) = 1/(x+1)

 

[Mark44]

The ln and exp functions are inverses of one another, so for any real number u, ln(e^u) = u. This means that you can simplify your function before taking its derivative. Then, what you end up differentiating is much simpler.