Practice section 4 of 8

Differentiate (a) $\cos^{-1}\sqrt{x}$ and (b) $\tan^{-1}(x^2)$ with respect to $x$.

Evaluate the first derivative of (a) $y=x^2$ when $x=2$, and (b) $y=\ln(2x^3)$ when $x=5$.

Let $f:x\mapsto1/x$. Using the first principles of differentiation, show that $f^{\prime}(x)=-1/x^2$.

Let $f:x\mapsto\sin x$. Using the first principles of differentiation, show that $f^{\prime}(x)=\cos x$.

Let $f:x\mapsto e^x$. Using the first principles of differentiation, show that $f^{\prime}(x)=e^x$.

Prove that $d(\cos^{-1}x)/dx=-1/\sqrt{1-x^2}$.