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

Question

Luke Alvarez · Accepted Answer

#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)#

ஔ டฟ 唔

Now that we have all the data required, we can 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)#

There's more simplification possible, but I'll let you handle the algebra.

I hope this is useful!

Elijah Abram · Answer

undefined To differentiate $ f(x) = x^7 \sqrt{4x^2 + 7} $ using the product rule, follow these steps:

1. Identify the two functions being multiplied: $ u(x) = x^7 $ and $ v(x) = \sqrt{4x^2 + 7} $.
2. Apply the product rule formula: $ (uv)' = u'v + uv' $.
3. Find the derivatives of $ u(x) $ and $ v(x) $ individually:
   - $ u'(x) = 7x^6 $ (using the power rule)
   - $ v'(x) = \frac{1}{2\sqrt{4x^2 + 7}} \cdot 8x $ (using the chain rule)
4. Substitute these derivatives into the product rule formula:
   - $ f'(x) = (7x^6)(\sqrt{4x^2 + 7}) + (x^7)\left(\frac{1}{2\sqrt{4x^2 + 7}} \cdot 8x\right) $
5. Simplify the expression:
   - $ f'(x) = 7x^6 \sqrt{4x^2 + 7} + 4x^8 $
   
So, the derivative of $ f(x) = x^7 \sqrt{4x^2 + 7} $ using the product rule is $ f'(x) = 7x^6 \sqrt{4x^2 + 7} + 4x^8 $.