Question

why does x^4+7 and x^4 have the same derivative, meaning can someone explain to me why they are both equal to the derivative 4x^3?

Answer 1

The derivative is SLOPE of the curve at x
It does not matter if you move the graph of the function up 7 units, the slope will be the same at each x. Graph it :)

Answer 2

a. Both have the exact same shape of curve, that is, except for position. Derivative means the function which defines the slope at any x, and both functions are the same.

go to your definition of derative.\

Lim d>0 of (f(x+d)-f(x))/d
= Lim d>0 of ((x+d)^4 -x^4)/d
= lim d>0 of (x^4+4x^3*d + ....-x^4)/d
= lim d>0 of (4x^3d + higher terms of d)/d
= 4x^3

Now do the same for f(x)=x^4+7
hint: the lim f(x+d)= (x+d)^4+7
and if you work that it shortly becomes the same as above.