Chain Rule

Question

Suppose that

f(x)=6x^2(4−4x)^3

Find an equation for the tangent line to the graph of f at x=−2, y=?

Answer 1

The function

The derivative at a point of a function of a single variable gives the slope of the tangent line

to the graph of the function at that point.

The line having this slope and passes through the point at gives the tangent line to f(x).

Now find the derivative of .

Apply the formula

Let .

At ,

This is the slope of the tangent line .

Now find the point on tangent line at .

Equation of line through the point and having slope m is .

So, the line through the point and having slope is

This is the tangent line.

The tangent line of the function is .

The value of y at is 41472.

The graph with tangent line is