How do you find instantaneous rate of change for the equation # y=4x^3+2x-3#?

The rate of change for an equation may simply be defined mathematically as the slope at any point.

if #f(x) = 4x^3 + 2x -3# then

#f'(x) = 12x^2 + 2# using the power rule from calculus.

For each polynomial term, multiply the exponent times the coefficient and decrease the exponent by one.

Thus for #f'(x), 4x^3# becomes #(3 xx 4) x^(3-1) = 12x^2# (power rule)

#f'(x)# for #2x# becomes #(1 xx 2)x^(1-1) = 2#

#f'(x)# for -3 becomes #(0 xx 3) = 0#

Combine the terms (sum rule): #12x^2 + 2#.