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Given y=3sec(2x), find (dy)/(dx). Answer: (dy)/(dx)=

Given y=3sec⁡(2x) y=3 \sec (2 x) , find dydx \frac{d y}{d x} .\newlineAnswer: dydx= \frac{d y}{d x}=

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Full solution

Q. Given y=3sec⁡(2x) y=3 \sec (2 x) , find dydx \frac{d y}{d x} .\newlineAnswer: dydx= \frac{d y}{d x}=
  1. Identify Function and Rules: Identify the function that needs to be differentiated and the rules that will be applied. The function is y=3sec⁡(2x)y = 3\sec(2x), and we will use the chain rule for differentiation, which 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.
  2. Differentiate Outer Function: Differentiate the outer function while keeping the inner function unchanged. The outer function is 3sec⁡(u)3\sec(u), where u=2xu = 2x. The derivative of sec⁡(u)\sec(u) with respect to uu is sec⁡(u)tan⁡(u)\sec(u)\tan(u), so the derivative of 3sec⁡(u)3\sec(u) with respect to uu is 3sec⁡(u)tan⁡(u)3\sec(u)\tan(u).
  3. Differentiate Inner Function: Differentiate the inner function. The inner function is u=2xu = 2x, and its derivative with respect to xx is dudx=2\frac{du}{dx} = 2.
  4. Apply Chain Rule: Apply the chain rule by multiplying the derivatives of the outer and inner functions. The derivative of yy with respect to xx is dydu⋅dudx=3sec⁡(u)tan⁡(u)⋅2\frac{dy}{du} \cdot \frac{du}{dx} = 3\sec(u)\tan(u) \cdot 2.
  5. Substitute Inner Function: Substitute the inner function back into the derivative to get the final answer. Replace uu with 2x2x to get (dydx)=3sec⁡(2x)tan⁡(2x)×2(\frac{dy}{dx}) = 3\sec(2x)\tan(2x) \times 2.
  6. Simplify Final Derivative: Simplify the expression to get the final derivative. The final derivative is (dydx)=6sec⁡(2x)tan⁡(2x)(\frac{dy}{dx}) = 6\sec(2x)\tan(2x).

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