Let *u* = 4*x*. Then we can rewrite: *e*^{4x} = *e*^{u}
One of the patterns says (*e*^{u})' = *e*^{u }⋅ *u*'. Since *u *= 4*x*, we know *u*' = 4. We substitute these into the pattern and get (*e*^{4x})' = (*e*^{u})' = *e*^{u} ⋅ *u*' = *e*^{4x }⋅ 4. We can also use these patterns to find antiderivatives. The equation (*e*^{u})' = *e*^{u} ⋅ *u*' means the derivative of *e*^{u} is *e*^{u} ⋅ *u*'. Thinking backwards, this means *e*^{u} is an antiderivative of *e*^{u} ⋅ *u*'. We can write the family of all antiderivatives of *e*^{u }⋅ *u*' as *e*^{u} + *C*.
Using indefinite integral notation, |