Question

Differentiate the following functions (i) 𝑓(𝑥) = 𝑥𝑒 𝑥 𝑐𝑠𝑐𝑥, (ii) ℎ(𝑥) = ln(𝑥 + √𝑥 2 − 1), (iii) 𝑦 = cos(𝑒 √𝑡𝑎𝑛 3𝑥 ), (iv) √1 + sin3(𝑥𝑦 2) = 𝑦, (v) 𝑦(𝑥) = (log𝑒 𝑥) 𝑥 tan(𝑒 𝑥 ), (vi) 𝑚(𝑥) = sin−1 (√1 − 9𝑥 2). 

Answer 1

i)

$f(x)=xe^xcscx$

$f'(x)=e^x(cscx-xcosx/sin^2x)+xe^xcscx$

ii)

$h(x)=ln(x+\sqrt{x^2-1})$

$h'(x)=\frac{1+x/\sqrt{x^2-1}}{x+\sqrt{x^2-1}}$

iii)

$y=cos(e^{\sqrt{tan(3x)}})$

$y'=-3sin(e^{\sqrt{tan(3x)}})sec^2(3x)e^{\sqrt{tan(3x)}}/(2\sqrt{tan(3x)})$

iv)

$y=\sqrt{1+sin^3(xy^2)}$

$y'=\frac{3sin^2(xy^2)cos(xy^2)(y^2+2xyy')}{2\sqrt{1+sin^3(xy^2)}}$

$y'=\frac{3sin^2(xy^2)cos(xy^2)y^2}{2\sqrt{1+sin^3(xy^2)}}/(1-\frac{3sin^2(xy^2)cos(xy^2)(y^2+2xy)}{2\sqrt{1+sin^3(xy^2)}})$

v)

$y(x)=xlog_extan(e^x)=xlnxtan(e^x)$

$y'(x)=tan(e^x)(1+lnx)+xe^xlnxsec^2(e^x)$

vi)

$m(x)=sin^{-1}(\sqrt{1-9x^2})$

$m'(x)=-\frac{18x}{6x\sqrt{1-9x^2}}=-\frac{3}{\sqrt{1-9x^2}}$