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Find the Derivative











































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

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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


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