How do you differentiate #f(x)=x^7sqrt(4x^2+7)# using the product rule?

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Answer 1 · 2016-09-01T15:13:51

#f'(x) = 7x^6(4x^2 + 7)^(1/2) + (4x^8)/(4x^2 + 7)^(1/2)#

Let #f(x) = g(x) xx h(x)#

The derivative of #f(x)#, by the product rule, is given by #f'(x) = g'(x) xx h(x) + g(x) xx h'(x)#

We must therefore differentiate both #g(x) and h(x)#.

#g'(x) = 7 xx x^(7 - 1)#

#g'(x) = 7x^6#

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

We differentiate #h(x)# using the chain rule.

Letting #y = u^(1/2)# and #u =4x^2 + 7#

#dy/dx = 1/2u^(-1/2) xx 8x#

#dy/dx = (8x)/(2(4x^2 + 7)^(1/2))#

#dy/dx = (4x)/(4x^2 + 7)^(1/2)#

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We now have all the information we need to apply the product rule.

#f'(x) = 7x^6 xx (4x^2 + 7)^(1/2) + (4x)/(4x^2 + 7)^(1/2) xx x^7#

#f'(x) = 7x^6(4x^2 + 7)^(1/2) + (4x^8)/(4x^2 + 7)^(1/2)#

This can be simplified further, but I'll leave the algebra to you.

Hopefully this helps!