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Let 
f(x)=sqrt(x^(3)).

f^(')(16)=

Let f(x)=x3 f(x)=\sqrt{x^{3}} .\newlinef(16)= f^{\prime}(16)=

Full solution

Q. Let f(x)=x3 f(x)=\sqrt{x^{3}} .\newlinef(16)= f^{\prime}(16)=
  1. Identify Function and Point: Identify the function and the point at which the derivative is to be evaluated.\newlineThe function given is f(x)=x3f(x) = \sqrt{x^3}, which can also be written as f(x)=(x3)1/2f(x) = (x^3)^{1/2}. We need to find the derivative of this function, denoted as f(x)f'(x), and then evaluate it at x=16x = 16.
  2. Apply Chain Rule: Differentiate the function using the chain rule.\newlineThe chain rule states that the derivative of a composite function is the derivative of the outer function evaluated at the inner function times the derivative of the inner function. In this case, the outer function is g(u)=u12g(u) = u^{\frac{1}{2}} and the inner function is u(x)=x3u(x) = x^3. Therefore, f(x)=g(u)u(x)f'(x) = g'(u) \cdot u'(x).
  3. Differentiate Outer Function: Differentiate the outer function g(u)=u12g(u) = u^{\frac{1}{2}} with respect to uu. The derivative of u12u^{\frac{1}{2}} with respect to uu is 12u12\frac{1}{2}u^{-\frac{1}{2}}.
  4. Differentiate Inner Function: Differentiate the inner function u(x)=x3u(x) = x^3 with respect to xx. The derivative of x3x^3 with respect to xx is 3x23x^2.
  5. Combine Derivatives: Combine the derivatives from Step 33 and Step 44 using the chain rule. f(x)=g(u)u(x)=12u123x2f'(x) = g'(u) \cdot u'(x) = \frac{1}{2}u^{-\frac{1}{2}} \cdot 3x^2. Now, substitute u=x3u = x^3 to get f(x)=12(x3)123x2f'(x) = \frac{1}{2}(x^3)^{-\frac{1}{2}} \cdot 3x^2.
  6. Simplify Expression: Simplify the expression for f(x)f'(x).f(x)=32x2(x3)12=32x2x32=32x232=32x12f'(x) = \frac{3}{2}x^2 \cdot (x^3)^{-\frac{1}{2}} = \frac{3}{2}x^2 \cdot x^{-\frac{3}{2}} = \frac{3}{2}x^{2 - \frac{3}{2}} = \frac{3}{2}x^{\frac{1}{2}}.
  7. Evaluate at x=16x = 16: Evaluate the derivative at x=16x = 16.f(16)=(32)(16)12=(32)(4)=3×2=6.f'(16) = \left(\frac{3}{2}\right)(16)^{\frac{1}{2}} = \left(\frac{3}{2}\right)(4) = 3 \times 2 = 6.

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