Discussion questions

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 $x>0$.

Find $\dfrac d{dx}[\sin^{-1}\sqrt{1-x^2}]$.

(a) Differentiate the following with respect to $x$, giving your answers as single fractions.

(i) $\ln\left(\dfrac{x}{\sqrt{1-2x}}\right)$,

(ii) $\dfrac1{\cos^{-1}(x^2)}$.

(b) The variables $x$ and $y$ are related by $e^{xy^2}=y(x^2+2e^x)$.

Find the value of $\dfrac{dy}{dx}$ when $x=0$ and $y=\dfrac12$.