Question

What is the slope of the curve at `t=3` assuming that the equations define x and y implicitly as differentiable functions `x=f(t)`, `y=g(t)`, and `x=t^5+t`, `y+4t^5=4x+t^4`?

Answer 1

If:
`x=t^5+t`
`y+4t^5=4(t^5+t)+t^4`
`y=-4t^5+4t^5+4t+t^4=t^4+4t`

Now:
`dx/dt=5t^4+1`
`dy/dt=4t^3+4`

But:
`dy/dx=dy/dtdt/dx`

So:
`dy/dx=(4t^3+4)/(5t^4+1)`

For `t=3` the slope will be:

`dy/dx=(4*(27+4))/(5*(81+1))=124/410=62/205=3.31`