Recitation 7

Exercise: Find the derivative of the function. f(z)=\frac{1}{z^2+1}.

By the quotient rule, f'(z)=\frac{-2z}{(z^2+1)^2}

Exercise: Find the first and second derivatives of the function. y=\cos(x^2).

By the chain rule, y'=-2x\sin(x^2). Again, by the product rule and the chain rule, y''=-2\sin(x^2)-4x^2\cos(x^2).

Exercise: Find all points on the graph of the function f(x)=2\sin x+\sin^2x at which the tangent line is horizontal.

Calculate the derivative, f'(x)=2\cos x+2\sin x\cos x=2\cos x(1+\sin x). The points at which the tangent line is horizontal are those whose derivatives are zero. Solve f'(x)=0, we get x=(k+\frac{1}{2})\pi, where k\in \mathbb{Z}.

Exercise: Find dy/dx by implicit differentiation. y\cos x=x^2+y^2.

Take derivative on both sides, we have y'\cos x-y\sin x=2x+2yy'. Hence y'=\frac{2x+y\sin x}{\cos x-2y}

Exercise: If f(x)+x^2[f(x)]^3=10 and f(1)=2, find f'(1).

Take derivative on both sides, we have f'(x)+2x[f(x)]^3+3x^2[f(x)]^2f'(x)=0. Let x=1. Get f'(1)=-\frac{16}{25}.

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