How do you differentiate #f(x) = (−7 x^2 − 5)^8 (2 x^2 − 9)^9# using the chain rule?

1 Answer
Jim H
Nov 27, 2016

#f(x) = (−7 x^2 − 5)^8 (2 x^2 − 9)^9#

# = (-1)^8 (7 x^2 + 5)^8 (2 x^2 − 9)^9#

# = (7 x^2 + 5)^8 (2 x^2 − 9)^9#

Use the product rule and the chain rule.

#f'(x) = [8 (7 x^2 + 5)^7 (14x)] (2 x^2 − 9)^9 + (7 x^2 + 5)^8 [9(2 x^2 − 9)^8 (4x)]#

Clean up the answer using algebra.

# = 112x(7 x^2 + 5)^7(2 x^2 − 9)^9 + 36x(7 x^2 + 5)^8(2 x^2 − 9)^8#

# = 4x(7 x^2 + 5)^7(2 x^2 − 9)^8[28(2x^2-9)+9(7x^2+5)]#

# = 4x(7 x^2 + 5)^7(2 x^2 − 9)^8[56x^2-252+63x^2+45]#

# = 4x(7 x^2 + 5)^7(2 x^2 − 9)^8(119x^2-207)#

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