# Derivatives

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# $\mathrm{f}\text{'}\left(\mathrm{x}\right)=\mathrm{\varphi }\left(\mathrm{x}\right) \mathrm{and} \mathrm{\varphi }\text{'}\left(\mathrm{x}\right)=\mathrm{f}\left(\mathrm{x}\right) \mathrm{for} \mathrm{all} \mathrm{x}. \mathrm{Also} \mathrm{f}\left(3\right)=5 \mathrm{and} \mathrm{f}\text{'}\left(3\right)=4.$ Then the value of ${\left[\mathrm{f}\left(10\right)\right]}^{2}-{\left[\mathrm{\varphi }\left(10\right)\right]}^{2}$ is _____

Moderate
Solution

## $\begin{array}{l}\frac{\mathrm{d}}{\mathrm{dx}}\left\{{\left[\left(\mathrm{f}\left(\mathrm{x}\right)\right)\right]}^{2}-{\left[\mathrm{\varphi }\left(\mathrm{x}\right)\right]}^{2}\right\}\\ =\left[\mathrm{f}\left(\mathrm{x}\right).\mathrm{f}\text{'}\left(\mathrm{x}\right)-\mathrm{\varphi }\left(\mathrm{x}\right)-\mathrm{\varphi }\text{'}\left(\mathrm{x}\right)\right]\\ =2\left[\mathrm{f}\left(\mathrm{x}\right).\mathrm{\varphi }\left(\mathrm{x}\right)-\mathrm{\varphi }\left(\mathrm{x}\right).\mathrm{f}\left(\mathrm{x}\right)\right] \left[ \mathrm{f}\text{'}\left(\mathrm{x}\right)=\mathrm{\varphi }\left(\mathrm{x}\right) \mathrm{and} \mathrm{\varphi }\text{'}\mathrm{x}=\mathrm{f}\left(\mathrm{x}\right)\right]\\ =0\\ \mathrm{or} {\left[\mathrm{f}\left(\mathrm{x}\right)\right]}^{2}-{\left[\mathrm{\varphi }\left(10\right)\right]}^{2}= \mathrm{constant}\\ \therefore {\left[\mathrm{f}\left(10\right)\right]}^{2}-{\left[\mathrm{\varphi }\left(\mathrm{x}\right)\right]}^{2}={\left[\mathrm{f}\left(3\right)\right]}^{2}-{\left[\mathrm{\varphi }\left(3\right)\right]}^{2}={\left[\mathrm{f}\left(3\right)\right]}^{2}-{\left[\mathrm{f}\text{'}\left(3\right)\right]}^{2}\\ =25-16=9\end{array}$

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If y = f(x) is an odd differentiable function defined on $\left(-\infty ,\infty \right)$ such that $\mathrm{f}\text{'}\left(3\right)=-2$ then $\mathrm{f}\text{'}\left(-3\right)$ equals ____