Identify Function and Point: Identify the function and the point at which the derivative is to be evaluated.The function given is f(x)=x3, which can also be written as f(x)=(x3)1/2. We need to find the derivative of this function, denoted as f′(x), and then evaluate it at x=16.
Apply Chain Rule: Differentiate the function using the chain rule.The 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)=u21 and the inner function is u(x)=x3. Therefore, f′(x)=g′(u)⋅u′(x).
Differentiate Outer Function: Differentiate the outer function g(u)=u21 with respect to u. The derivative of u21 with respect to u is 21u−21.
Differentiate Inner Function: Differentiate the inner function u(x)=x3 with respect to x. The derivative of x3 with respect to x is 3x2.
Combine Derivatives: Combine the derivatives from Step 3 and Step 4 using the chain rule. f′(x)=g′(u)⋅u′(x)=21u−21⋅3x2. Now, substitute u=x3 to get f′(x)=21(x3)−21⋅3x2.
Simplify Expression: Simplify the expression for f′(x).f′(x)=23x2⋅(x3)−21=23x2⋅x−23=23x2−23=23x21.
Evaluate at x=16: Evaluate the derivative at x=16.f′(16)=(23)(16)21=(23)(4)=3×2=6.
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