Find the derivative of (x^3)√[(x^2) - 8]

Question

I know how to solve derivatives, but this one is just a bit daunting and I don't know where to start with it.

Let f(x) = x³ √(x² – 8), find f´(x).

Answer 1

Here is the link to the Desmos graph referenced in the video:

https://www.desmos.com/calculator/orbdzqz2dn

Be sure to pause the video and try to work out the problem yourself after you see the initial suggestions.

Answer 2

f(x) = g(x)*h(x) where g(x) = x^3 and h(x) = sqrt(x^2 - 8) = (x^2 - 8)^(1/2)

So you need the product rule:

f(x) = g(x)*h'x) + g'(x) * h(x)

g'(x) = 3*x^2 while h'(x) = (1/2)(x^2- 8)^(-1/2) (2x) = x(x^2-8)^(-1/2) <--- power rule and chain rule

The derivative is:

x^3 * (x)(x^2-8)^(-1/2) + (x^2-8)^(1/2) * (3x^2)

x^4(x^2-8)^(-1/2) + (3x^2)(x^2-8)^(1/2) <--- rule of exponents; commutative

(x^2-8)^(-1/2) ( x^4 - 3x^2(x^2-8)) = <--- factors out (x^2-8)^(-1/2)

Remember: m^(-1/2) - m^(1/2) = m^(-1/2) [ 1 - m]

you SUBTRACT the exponents when factoring

exponent 1/2 - -1/2 = 1/2+1/2 = 1

(x^2-8)^(-1/2) ( x^4 - 3x^4 - 24x^2)

(x^2 - 8)^(-1/2) ( -2x^4 - 24x^2)

(-2x^2)( (x^2 + 12)(x^2 - 8)^(-1/2)

OR

(-2x^2)(x^2 + 12)/ (x^2 - 8)^(1/2) =

(-2x^2)(x^2+12)/ sqrt(x^2 - 8)