Question #64342

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Answer 1 · 2017-07-30T00:16:57

#d/(dx) [3(sqrtx)^2 · (3x^2-1)+5x^3] = color(blue)(ul(42x^2 - 3#

We're asked to find the derivative

#d/(dx) [3(sqrtx)^2 · (3x^2-1)+5x^3]#

which is the same as

#d/(dx) [3x(3x^2-1)+5x^3]#

Using the power rule on #5x^3#, and factoring out the constant, #3#:

#= 15x^2 + 3d/(dx)[x(3x^2-1)]#

Using the product rule, which is

#d/(dx)[uv] = v(du)/(dx) + u(dv)/(dx)#

where

#u = x#

#v = 3x^2-1#:

#= 15x^2 + 3((3x^2-1)d/(dx)[x] + xd/(dx)[3x^2-1])#

The derivative of #x# is #1#, and the derivative of #3x^2-1# is #6x# (both from power rule):

#= 15x^2 + 3((3x^2-1)(1) + x(6x))#

#= 15x^2 + 9x^2 - 3 + 18x^2#

#= color(blue)(ul(42x^2-3#

At least two other calculators agree with me, so the given answer must be incorrect...maybe you were doing it right after all!