Q:
A. Implicitly find dy/dx of exy=8
B. Now, solve exy=8 for y first, then take the derivative. Compare your answers to A and B.
Answer:
A. Remember, implicit differentiation is just the chain rule: y is a function of x. So, we need to use the product rule since we are multiplying two functions ex times y:
Also remember, the derivative of x with respect to x is 1 and the derivative of y with respect to x is dy/dx:
Differentiate as follows:
exy + ex(dy/dx) = 0
Now, solve for dy/dx:
ex(dy/dx) = -exy
dy/dx = -exy/ex
dy/dx = -y [final answer]
However, we may want to put the answer in terms of x. If this is the case, we know from the original problem:
exy=8 which implies:
y = 8/ex
So, dy/dx = -y = -8/ex
B. Solve for y first, then take the derivative:
We just solved for y above and got:
y = 8/ex
We can rewrite this as:
y = 8*e-x
Now, take the derivative (using the chain rule)
dy/dx = 8*e-x*(-1)
dy/dx = -8*e-x
Which can then be re-written as:
dy/dx = -8/ex
Compare A and B? Notice they are identical — which is good news. Whether we solve for y first or differentiate implicitly, we still get the same answer for dy/dx.
so much help! thank you!
now can you show me how to do it like this?
A. find dy/dx using implicit differentiation lnx/y=6-x ?
B. compare your answer with the result obtained by first solving for y as a function of x and then taking the derivative
LikeLike