Let (y = ^ 3 ! (e^ ,2x^ 2 +1 ) ); that is, (y ) is the cube of the sine of an…

MATHEMATICS · Multi-part · A Level · Mathematics · Differentiation · Differentiation

Problem

Let y=sin⁡3 ⁣(e 2x2+1)y = \sin^{3}\!\bigl(e^{\,2x^{2}+1}\bigr); that is, yy is the cube of the sine of an exponential function. (a) Derive an unsimplified expression for dydx\dfrac{dy}{dx} by differentiating step by step from the outermost to the innermost function. (b) Algebraically simplify your result from part (a) so that no trigonometric or exponential factor is unnecessarily repeated. Express the final answer as a single product of elementary factors. (c) Evaluate your simplified derivative at x=0x = 0, giving an exact value.

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