Can you help me find the derivative of f(x) = (4x^3 +1) * sin -1^(x^3)?

What is the derivative of (4x^3 +1)?
What is the derivative of sin^-1(x^3)?
Now use the product rule.
Please show your work so we can help you.
 
Steven G said:
What is the derivative of (4x^3 +1)?
What is the derivative of sin^-1(x^3)?
Now use the product rule.
Click to expand...
Hi Steven. I'm not sure what's up with dboy; they haven't responded in either of their threads.

For future readers:

The 1st derivative of 4x^3+1 is found term-by-term, using the Power Rule.

d/dx [ 4x^3 + 1 ] = 3 ∙ 4x^(3–1) + 0 = 12x^2

The 1st derivative of arcsin(x^3) is found using the arcsine derivation and the chain rule (because arcsine of x^3 is a function composition – that is, a function within a function).

d/dx [ arcsin(x) ] = 1/√(1 – x^2)

Therefore,

d/dx [arcsin(x^3)] = 1/√(1 – [x^3]^2) ∙ 3 ∙ x^(3 – 2) = 3x^2/√(1 – x^6)

The function f(x) in this thread is a product of a polynomial factor (4x^3+1) and a trigonometric factor (arcsin(x^3)). The Product Rule tells us to (1) multiply the polynomial factor by the derivative of the trig factor, (2) multiply the trig factor by the derivative of the polynomial factor and then (3) add those two results.

d/dx [ (4x^3 + 1) ∙ arcsin(x^3) ] = (4x^3 + 1) ∙ 3x^2/√(1 – x^6) + arcsin(x^3) ∙ (12x^2)

The exercise answer is [imath]\frac{(3x^2)(4x^3 \,+\, 1)}{\sqrt{1 \;–\; x^6}} + 12x^2 \cdot \arcsin(x^3)[/imath] :)
[imath]\;[/imath]
 
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