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h(x)=(arctan(x^(5)+2x))

h(x)=(arctan⁡(x5+2x)) h(x)=\left(\arctan \left(x^{5}+2 x\right)\right)

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Full solution

Q. h(x)=(arctan⁡(x5+2x)) h(x)=\left(\arctan \left(x^{5}+2 x\right)\right)
  1. Identify Functions: Identify the outer function and the inner function for the chain rule.\newlineOuter function: arctan⁡(u)\arctan(u), Inner function: u=x5+2xu = x^5 + 2x.
  2. Differentiate Outer Function: Differentiate the outer function with respect to the inner function uu.ddx(arctan⁡(u))=11+u2\frac{d}{dx}(\arctan(u)) = \frac{1}{1+u^2}.
  3. Differentiate Inner Function: Differentiate the inner function with respect to xx.ddx(x5+2x)=5x4+2.\frac{d}{dx}(x^5 + 2x) = 5x^4 + 2.
  4. Apply Chain Rule: Apply the chain rule: the derivative of the outer function times the derivative of the inner function.\newlineh′(x)=11+(x5+2x)2⋅(5x4+2)h'(x) = \frac{1}{1+(x^5 + 2x)^2} \cdot (5x^4 + 2).
  5. Simplify Expression: Simplify the expression if possible.\newlineh′(x)=5x4+21+(x10+4x6+4x2)h'(x) = \frac{5x^4 + 2}{1 + (x^{10} + 4x^6 + 4x^2)}.

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