Suppose that f(x) = 5x(1-5x)^4.

Question

Find an equation for the tangent line to the graph of f at x=1.

Tangent line: y = ?

Answer 1

To find the equation for the tangent line, you first find the derivative of f(x), which is f'(x).

So, f'(x)=5*(5x-1)^3*(25x-1).

Then, you plug into f'(x) the value given for x, in this case it is 1.

f'(1)=5*(5-1)^3*(25-1)=7680. This number, 7680, is the slope of the tangent line.

All you need now is the point on the tangent line.

You plug in the value of x given into the original equation:

f(1)=5*(1-5)^4=1280.

Now, we have the points (1,1280), we could use this which is in the form (x1,y1) and the slope, 7680, which is 'm.' You then plug into the point-slope formula.

y-y1=m(x-x1)

y-1280=7680(x-1)

which simplifies to

y=7680x-6400