Practice section 5 of 8

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

Evaluate $\left.\dfrac{d}{dx}\,2^{3x^2+\sin^2(e^{x^2})}\right|_{x=1}$, correct to 5 significant figures.

Find the derivatives of the following

(a) $\dfrac{\pi\sqrt{8100+x^2}}x$,

(b) $\sec^3(2x)$,

(c) $\sin\sqrt{x^2+1}$,

(d) $\log_{10}5x$,

(e) $\ln[(x^3+2)(x^2+3)]$,

(f) $x^2e^{x\sin2x}$,

(g) $\left(\dfrac1{\sin^{-1}3x}\right)^4$,

(h) $\sin^{-1}(2\sin x)$,

(i) $\tan^{-1}\left(\dfrac{1+x}{1-x}\right)$.

A curve has equation $xy^2+3e^y=4x$. Find $\dfrac{dy}{dx}$ in terms of $x$ and $y$.

A curve has equation $\sin^{-1}y+xe^y=3x$. Find $\dfrac{dy}{dx}$ in terms of $x$ and $y$.

Find $\dfrac{dy}{dx}$ given that $y=x^{\sqrt{x}}$ where $x>0$.