Find the implicit differentiation, d/dx tan(5x + y) = 5x
You want to perform implicit differentiation on the expression: tan(5x + y) = 5x.
We will be differentiating both sides (implicitly) wrt x.
The differential (implicit) of the rhs side is d(5x) / dx = 5.
We now use the chain rule to differentiate the lhs.
Let u = 5x + y,
With du / dx = 5 + dy / dx
Then lhs = tan(u),
And d(lhs) / dx = d(lhs) / du * du / dx = sec^2(u) * (5 + dy / dx).
i.e. sec^2(u) * (5 + dy / dx) = 5
sec^2(5x + y) * (5 + y’) = 5
(5 + y’) = 5/(sec^2(5x + y) )
y’ = 5/(sec^2(5x + y) ) – 5
y’ = 5{1/(sec^2(5x + y) ) – 1}