f(x)= 3x^7 +2x -9
If g(x)= f inverse of x
and g(-4)= 1,
what is g'(-4)?
I know that f inverse is the same as
1/f'(x) but I'm not sure where to go from there.
To find g'(-4), we need to first find the derivative of f(x), and then use that information to find the derivative of g(x) at x = -4.
Let's start by finding the derivative of f(x):
f(x) = 3x^7 + 2x - 9
To find the derivative, we take the derivative of each term separately:
The derivative of 3x^7 is 21x^6, since the power rule for differentiation states that the derivative of x^n is n*x^(n-1).
The derivative of 2x is 2, since the derivative of a constant multiplied by x is just the constant.
The derivative of -9 is 0, since the derivative of a constant is always zero.
Now we can write the derivative of f(x):
f'(x) = 21x^6 + 2
Next, we can find the value of g(-4) = 1. This means that when x = -4, g(x) = 1.
To find g'(-4), we need to evaluate the derivative of g(x) at x = -4. Since g(x) is the inverse of f(x), we can use the fact that the derivative of an inverse function is the reciprocal of the derivative of the original function at the corresponding point.
Therefore, we have:
g'(-4) = 1 / f'(-4)
To find f'(-4), we need to substitute -4 into the derivative expression we found earlier:
f'(-4) = 21(-4)^6 + 2
f'(-4) = 21(4096) + 2
Calculating this expression, we get:
f'(-4) = 86018
Finally, we can substitute this value into the equation for g'(-4):
g'(-4) = 1 / 86018
So, g'(-4) is equal to 1 divided by 86018.