What is the derivative of #f(x) = (−7 x^2 − 5)^8 * (2 x^2 − 9)^9#?

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Answer 1 · 2015-07-29T12:59:38

The given function is the product of two functions and the derivative may be computed using he product rule.

Let, #g(x) = ( -7x^2 - 5)^8# and #h(x) = (2x^2 - 9)^9#

Thus, #f(x) = g(x)*h(x)#

Now, #(df)/(dx) = g*(dh)/(dx) + h*(dg)/(dx) #

Now,
#(dh)/(dx) = 9(2x^2 - 9)^(9-1)*(d(2x^2 - 9))/(dx)#

# = 9(2x^2 - 9)^8*(4x)#

Also,
#(dg)/(dx) = 8( -7x^2 - 5)^(8-1)*(d( -7x^2 - 5))/(dx)#

#= 8( -7x^2 - 5)^7*(-14x)#

Now do rearrange the terms. We already know #h(x), g(x)# and their derivatives #(dh)/(dx)# and #(dg)/(dx)#.