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Find the derivative of f(x) f(x) . \newlinef(x)=x+3 f(x) = \sqrt{x+3} \newlinef′(x)= f'(x) = ______

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Q. Find the derivative of f(x) f(x) . \newlinef(x)=x+3 f(x) = \sqrt{x+3} \newlinef′(x)= f'(x) = ______
  1. Identify Functions: Identify the inner and outer functions for the composition.\newlineInner function, u(x)=(x+3)u(x) = (x + 3).\newlineOuter function, f(u)=uf(u) = \sqrt{u}, where uu is (x+3)(x + 3).
  2. Derivative of Outer Function: Determine the derivative of the outer function, f(u)=uf(u) = \sqrt{u}.\newlineUsing the power rule, the derivative of u12u^{\frac{1}{2}} is (12)u−12\left(\frac{1}{2}\right)u^{-\frac{1}{2}}.\newlineTherefore, f′(u)=12uf'(u) = \frac{1}{2\sqrt{u}}.
  3. Substitute into Derivative: Substitute u(x)=(x+3)u(x) = (x + 3) into the derivative of the outer function.f′(u(x))=12u(x)=12x+3.f'(u(x)) = \frac{1}{2\sqrt{u(x)}} = \frac{1}{2\sqrt{x+3}}.
  4. Derivative of Inner Function: Find the derivative of the inner function, u(x)=x+3u(x) = x + 3.\newlineThe derivative of xx is 11, and the derivative of a constant is 00.\newlineTherefore, u′(x)=1+0=1u'(x) = 1 + 0 = 1.
  5. Apply Chain Rule: Apply the chain rule: ddx[f(u(x))]=f′(u(x))⋅u′(x)\frac{d}{dx} [f(u(x))] = f'(u(x)) \cdot u'(x).f′(u(x))=12x+3f'(u(x)) = \frac{1}{2\sqrt{x+3}} and u′(x)=1u'(x) = 1. Therefore, f′(x)=12x+3⋅1=12x+3f'(x) = \frac{1}{2\sqrt{x+3}} \cdot 1 = \frac{1}{2\sqrt{x+3}}.

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