Calculus question

Question

Why is the derivative of x^3 equal to 3x^2?

Shouldn't the derivative be a linear function or a constant since we're dealing with the slope of a tangent line and why is the derivative a quadratic like 3x^2.

Answer 1

I think what you are confused about is that the general slope for every point on the cubic curve, f(x) = x³ is 3x². For the slope of a tangent line at a given point, you have to evaluate 3x² at a given point.

For example to find the tangent line of a curve at (2, 8), the slope at that point, m = 3(2)² = 12.

The equation for the line tangent to x³ at (2,8) is:

y-y₀ = m(x-x₀)

y-8 = 12(x-2) = 12x-24

y = 12x - 16

If you plot the two lines, y=x³ and y=12x-24 with something like Desmos, you will see that the tangent line just touches the cubic curve at (2,8).

Answer 2

the slope is a constant if the graph is a linear equation, with x^1,

the slope of a quadratic, x^2 = 2x

the slope of any polynomial has highest degree one less than the degree of the polynomial

slope of x^4 is 4x^3

slope of x^3 is 3x^2

slope of x^2 is 2x

slope of x is 1

slope of 1 is 0

slope of x^-1 is -x^-2

slope of x^-2 is -x^-3

slope of x^-3 is -x^-4

slope of lnx = 1/x, but that's not a polynomial with integer exponents

the slope is the limit as h approaches zero of [f(x+h)-f(x)]/h

use that definition to calculate the derivative of x^3 and x^2

that's the slope of a secant line as the secant line approaches the tangent line

the tangent line is the limit as the secant line approaches zero.