Find the derivative of the following function:

f(x)=(1x2+1x3)(x3.5+x2.5)

Question

Find the derivative of the following function:

f(x)=(1x2+1x3)(x3.5+x2.5)

BlackTom AI · Accepted Answer

The question asks for the derivative of f(x) where f(x) = (x^2 + x^3)(x^3.5 + x^2.5).

First, identify the structure: f is a product of two functions, u(x) = x^2 + x^3 and v(x) = x^3.5 + x^2.5. We will apply the product rule: f'(x) = u'(x)·v(x) + u(x)·v'(x).

Compute each derivative:
- u'(x) = d/dx (x^2 + x^3) = 2x + 3x^2.
- v'(x) = d/dx (x^3.5 + x^2.5) = 3.5·x^(3.5−1) + 2.5·x^(2.5−1) = 3.5x^2.5 + 2.5x^1.5.
  Note: x^3.5 is x^(7/2) and its derivative is (7/2)x^(5/2) = 3.5x^2.5; similarly x^2.5 is x^(5/2) with derivative (5/2)x^(3/2) = 2.5x^1.5.

Assemble f'(x) using the product rule:
f'(x) = (2x + 3x^2)(x^3.5 + x^2.5) + (x^2 + x^3)(3.5x^2.5 + 2.5x^1.5).

If you wanted to factor or simplify further, you could factor common powers of x from each term, for example pulling x^1.5 or x^2.5 factors, but the above directly applies the product and chain rules and is a correct expression for the derivative.

Note about the answer options: the provided options are missing in the prompt, so a direct comparison cannot be performed. When options are available, compare each candidate against this derivative form, checking for correct application of the product rule and accurate exponents (including fractional powers).

ECON5005 (ND) Quiz 2 -2025 S1

Find the derivative of the following function: f(x)=(1x2+1x3)(x3.5+x2.5)

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