# Find the Derivative

what is the derivative of the function y=3(8-x^2)^6

### Comments for Find the Derivative

 Oct 19, 2011 Find the Derivative by: Staff Question: what is the derivative of the function y=3(8-x^2)^6 Answer: Function f(x) = 3 * (8 - x^2)^6 take the derivative d/dx[3 * (8 - x^2)^6] use the derivative power rule for the problem as a whole: d(Uⁿ) = nUⁿ⁻¹dU part A: consider only this part of the function using the Power Rule: x^2 d/dx[x^2 ] = 2 * x^(2 - 1) * 1 part B: consider this part of the function using the Subtraction Rule: 8 - x^2 d/dx[8-x^2] = 0 - 2*x^(2 - 1) * 1 part C: apply the Power Rule d(Uⁿ) = nUⁿ⁻¹dU to this part of the function: (8 - x^2)^6 d/dx[(8 - x^2)^6] = [6 * (8 - x^2)^(6 - 1)] * [0 - 2 * x^(2 - 1) * 1] part D: apply the Variable rule: d(nU) = ndU: d/dx[3 * (8 - x^2)^6] = 3 * d/dx[(8 - x^2)^6] 3 * d/dx[(8 -x^2)^6] = 3 * {[6 * (8 - x^2)^(6-1)] * [0 - 2 * x^(2 - 1) * 1]} d/dx[3 * (8 - x^2)^6] = 3 * 6 * (8 - x^2)^(6 - 1) * [0 - 2*x^(2 - 1)*1] part E: simplify the result d/dx[3 * (8 - x^2)^6] = -18 * (8 - x^2)^(6 - 1) * 2 * x^(2 - 1) d/dx[3 * (8 -x^2)^6] = -18 * (8 - x^2)^5 * 2 * x^1 d/dx[3 * (8 - x^2)^6] = -18 * (8 - x^2)^5 * 2* x d/dx[3 * (8 - x^2)^6] = -36 * x * (8 - x^2)^5 d/dx[3*(8 - x^2)^6] = -36x(8 - x^2)^5 the final answer is: d/dx[ 3(8 - x^2)^6] = -36x(8 - x^2)^5 if you expand the expression d/dx[ 3(8 - x^2)^6] = -36x(8 - x^2)^5 = 36x^11 - 1440x^9 + 23040x^7 - 184320x^5 + 737280x^3 - 1179648x Or, you can always expand the expression first, and then take the derivative of the expanded expression: f(x) = 3*(8 - x^2)^6 f(x) = 3 * (x^12 - 48x^10 + 960x^8 - 10240x^6 + 61440x^4 - 196608x^2 + 262144) f(x) = 3x^12 - 144x^10 + 2880x^8 - 30720x^6 + 184320x^4 - 589824x^2 + 786432 f’(x) = 36*x^11 - 1440*x^9 + 23040*x^7 - 184320*x^5 + 737280*x^3 - 1179648*x f’(x) = 36x^11 - 1440x^9 + 23040x^7 - 184320x^5 + 737280x^3 - 1179648x Thanks for writing. Staff www.solving-math-problems.com