Find the slope and equation of the tangent line to the graph of the function at the given value of x.

f(x)=x^4-20x^2+64;x=-1

f'(x) = 4x^3-40x

f'(-1) = -4+40 = 36

f(-1) = 45
so the line is
y-45 = 36(x+1)

To find the slope and equation of the tangent line to the graph of a function at a given value of x, you need to take the derivative of the function and evaluate it at the given value of x.

Let's start by finding the derivative of the function f(x) = x^4 - 20x^2 + 64. The derivative of a function represents the rate of change of the function at any given point. In this case, it will give us the slope of the tangent line.

Using the power rule for differentiation, we can differentiate each term of the function separately. The power rule states that if we have a term of the form x^n, the derivative is nx^(n-1).

For the first term, x^4, the derivative is:
d/dx (x^4) = 4x^3

For the second term, -20x^2, the derivative is:
d/dx (-20x^2) = -40x

And for the constant term, 64, the derivative is zero, since a constant has a derivative of zero.

Now, we can find the derivative of the entire function by adding up the derivatives of each term:
f'(x) = 4x^3 - 40x

To find the slope of the tangent line at x = -1, we need to evaluate the derivative at that point:
f'(-1) = 4(-1)^3 - 40(-1)
= 4(-1) - 40(-1)
= -4 +40
= 36

So, the slope of the tangent line is 36.

To find the equation of the tangent line, we need to use the point-slope form of a linear equation, which is y - y1 = m(x - x1), where (x1, y1) is a point on the line, and m is the slope.

We already have the slope, which is 36, and the given point is (-1, f(-1)). To find the y-coordinate of the point, we need to substitute x = -1 into the original function f(x):
f(-1) = (-1)^4 - 20(-1)^2 + 64
= 1 - 20 + 64
= 45

So, the point is (-1, 45).

Substituting the values into the point-slope form equation, we get:
y - 45 = 36(x - (-1))
y - 45 = 36(x + 1)
y - 45 = 36x + 36
y = 36x + 81

Therefore, the equation of the tangent line to the graph of the function f(x) = x^4 - 20x^2 + 64 at x = -1 is y = 36x + 81.