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 $f'(1)=-\frac{16}{25}$.