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Find f^(')(theta) if f(theta)=(7)/(5)sin^(2)(2theta)

Find f′(θ) f^{\prime}(\theta) if f(θ)=75sin⁡2(2θ) f(\theta)=\frac{7}{5} \sin ^{2}(2 \theta)

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Full solution

Q. Find f′(θ) f^{\prime}(\theta) if f(θ)=75sin⁡2(2θ) f(\theta)=\frac{7}{5} \sin ^{2}(2 \theta)
  1. Identify Function: Identify the function to differentiate. f(θ)=(75)sin⁡2(2θ) f(\theta) = \left(\frac{7}{5}\right)\sin^2(2\theta)
  2. Apply Chain Rule: Apply the chain rule for differentiation: ddθ[sin⁡2(2θ)]=2sin⁡(2θ)cos⁡(2θ)⋅ddθ[2θ]\frac{d}{d\theta} [\sin^2(2\theta)] = 2\sin(2\theta)\cos(2\theta) \cdot \frac{d}{d\theta} [2\theta]
  3. Differentiate Inner Function: Differentiate the inner function 2θ2\theta.ddθ[2θ]=2\frac{d}{d\theta} [2\theta] = 2
  4. Substitute Back: Substitute back to get f′(θ)f'(\theta).
    f′(θ)=75×2×sin⁡(2θ)×cos⁡(2θ)×2f'(\theta) = \frac{7}{5} \times 2 \times \sin(2\theta) \times \cos(2\theta) \times 2
  5. Simplify Expression: Simplify the expression. f′(θ)=(285)sin⁡(2θ)cos⁡(2θ) f'(\theta) = \left(\frac{28}{5}\right)\sin(2\theta)\cos(2\theta)

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