Assuming that the equations define x and y implicitly as differentiable functions x = f(t), y = g(t), find the slope of the curve x = f(t),

y = g(t) at the given value of t.

i tried to differential both functions but there are still y and x-variables. can someone guide me through this?

This is okay, consider the following

\(\displaystyle y=f(x)= e^{x+5}\)

And

\(\displaystyle lny = lne^{x+5} = x+5 \)

If we want to differentiate the above implicitly we get,

\(\displaystyle \frac{1}{y} y^{ \prime } = 1 \)

Which follows

\(\displaystyle y^{ \prime} = y \)

Notice how \(\displaystyle y^{ \prime} = y = f(x) \)

This is okay!

\(\displaystyle y^{ \prime} = y = e^{x+5}\)

We are allowed to have our derivative as a function of our original function, there is no problem.

Knowing this can you now compute your question?