Home > Calculus > Product Rule

What is the derivative of #x^3 * (2/3x^2 -1)^4#?

Answer 1

Paisley Johnson

#176/81 x^10 -32/3 x^8 +56/3 x^6 -40/3 x^4 +3x^2#

Extend the phrase as

#x^3 [ (2/3 x^2)^4 +4 (2/3 x^2)^3 (-1) +6 (2/3 x^2)^2 (-1)^2 +4(2/3 x^2) (-1)^3 +(-1)^4]#

=#16/81 x^11 -32/27 x^9 +24/9 x^7 -8/3 x^5 +x^3#

The first derivative of it would be

#176/81 x^10 -32/3 x^8 +56/3 x^6 -40/3 x^4 +3x^2#

The product rule of differentiation is an additional option.

By signing up, you agree to our Terms of Service and Privacy Policy

Answer 2

Emma Aldridge

To find the derivative of the function (f(x) = x^3 \cdot \left(\frac{2}{3}x^2 - 1\right)^4), you can use the product rule and chain rule. The derivative is:

[f'(x) = 3x^2 \cdot \left(\frac{2}{3}x^2 - 1\right)^4 + x^3 \cdot 4\left(\frac{2}{3}x^2 - 1\right)^3 \cdot \frac{4}{3}x]

By signing up, you agree to our Terms of Service and Privacy Policy

Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.