Question

1. y = x^3 - 6x^2 find tangent line

Explanation:

Step1: Find the derivative

The derivative of $y = x^{3}-6x^{2}$ using the power - rule $\frac{d}{dx}(x^{n})=nx^{n - 1}$ is $y'=3x^{2}-12x$.

Step2: General form of tangent line

The equation of a tangent line to the curve $y = f(x)$ at the point $(x_0,y_0)$ is $y - y_0=y'(x_0)(x - x_0)$. Since no specific point $(x_0,y_0)$ is given, if we assume the point of tangency is $(x_0,x_0^{3}-6x_0^{2})$, the slope of the tangent line at that point is $m = 3x_0^{2}-12x_0$. Then the equation of the tangent line is $y-(x_0^{3}-6x_0^{2})=(3x_0^{2}-12x_0)(x - x_0)$.
Expanding gives $y-(x_0^{3}-6x_0^{2})=3x_0^{2}x-3x_0^{3}-12x_0x + 12x_0^{2}$, and $y=3x_0^{2}x-3x_0^{3}-12x_0x + 12x_0^{2}+x_0^{3}-6x_0^{2}$, so $y=(3x_0^{2}-12x_0)x - 2x_0^{3}+6x_0^{2}$.

Answer:

The equation of the tangent line to the curve $y = x^{3}-6x^{2}$ at the point $(x_0,x_0^{3}-6x_0^{2})$ is $y=(3x_0^{2}-12x_0)x - 2x_0^{3}+6x_0^{2}$.