Suppose that f(x)=6x^−4+9x^−2. Evaluate each of the following:
f'(3)=
f'(-1)
To evaluate f'(x), the derivative of the function f(x), we can use the power rule of differentiation. The power rule states that for a term of the form ax^n, the derivative is given by taking the exponent n, multiplying it by the coefficient a, and then reducing the exponent by 1.
Given f(x) = 6x^(-4) + 9x^(-2), we can find f'(x) by differentiating each term separately.
Term 1: 6x^(-4)
Applying the power rule, we get: [6 * (-4) * x^(-4-1)] = -24x^(-5)
Term 2: 9x^(-2)
Applying the power rule, we get: [9 * (-2) * x^(-2-1)] = -18x^(-3)
Now we can find f'(x) by adding the derivatives of each term:
f'(x) = -24x^(-5) - 18x^(-3)
To evaluate f'(3), we substitute x = 3 into the expression for f'(x):
f'(3) = -24(3)^(-5) - 18(3)^(-3)
= -24/3^5 - 18/3^3
= -24/243 - 18/27
= -24/243 - 18/27 * 3/3
= -24/243 - 54/243
= (-24 - 54)/243
= -78/243
= -26/81
Therefore, f'(3) = -26/81.
To evaluate f'(-1), we substitute x = -1 into the expression for f'(x):
f'(-1) = -24(-1)^(-5) - 18(-1)^(-3)
= -24/(-1)^5 - 18/(-1)^3
= -24/(-1) - 18/(-1)
= 24 - 18
= 6
Therefore, f'(-1) = 6.