Fallbasiss4

2022-01-23

The differential for each function can be found:
$\left(a\right)y={x}^{2}\mathrm{sin}\left(4x\right)$

$\left(b\right)y=\mathrm{ln}\left(\sqrt{\left(1+{t}^{2}\right)}\right)$

dodato0n

Expert

a) Given function is
$y={x}^{2}\mathrm{sin}4x$
Here, we use product rule of differentiation to find the differential
$dy=\left(\mathrm{sin}4x\right)d\left({x}^{2}\right)+{x}^{2}d\left(\mathrm{sin}4x\right)$
Differential of ${x}^{2}$ is $2xdx$ and that of $\mathrm{sin}4x$ is $4\mathrm{cos}4x$
$dy=\left(2x\mathrm{sin}4x\right)dx+\left(4{x}^{2}\mathrm{cos}4x\right)dx$
$dy=\left(2x\mathrm{sin}4x+4{x}^{2}\mathrm{cos}4x\right)$

Hana Larsen

Expert

b) Given function is
$y=\mathrm{ln}\sqrt{1+{t}^{2}}$
Firstly, we apply rule of logarithms ${\mathrm{log}m}^{n}=n\mathrm{log}m$
$y=\mathrm{ln}\sqrt{1+{t}^{2}}$
$y={\mathrm{ln}\left(1+{t}^{2}\right)}^{\frac{1}{2}}$
$y=\frac{1}{2}\mathrm{ln}\left(1+{t}^{2}\right)$
Differential of $\mathrm{log}x$ is $\frac{dx}{x}$ and that of ${x}^{2}$ is $2xdx$
$dy=\frac{1}{2}\frac{1}{\left(2+{t}^{2}\right)}d\left(1+{t}^{2}\right)$
$dy=\frac{1}{2\left(1+{t}^{2}\right)}2tdt$
$dy=\frac{t}{\left(1+{t}^{2}\right)}dt$

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