Differentiate `f(x)= (1/x)sinx` by using the definition of the derivative. Hints: You may use any of the following: `sin(x+h)=sinxcos h +...

Friday, March 21, 2014

Differentiate $\displaystyle{f{{\left({x}\right)}}}={\left(\frac{{1}}{{x}}\right)}{\sin{{x}}}$ by using the definition of the derivative. Hints: You may use any of the following: $\displaystyle{\sin{{\left({x}+{h}\right)}}}={\sin{{x}}}{\cos{{h}}}+\ldots$

Hello!

By the definition of the derivative we need to find the limit of $\displaystyle\frac{{{f{{\left({x}+{h}\right)}}}-{f{{\left({x}\right)}}}}}{{h}}$ for $\displaystyle{h}-\gt{0}$ and any fixed x. Consider the difference:

$\displaystyle{f{{\left({x}+{h}\right)}}}-{f{{\left({x}\right)}}}=\frac{{{\sin{{\left({x}+{h}\right)}}}}}{{{x}+{h}}}-\frac{{{\sin{{\left({x}\right)}}}}}{{x}}=$

$\displaystyle=\frac{{1}}{{{x}{\left({x}+{h}\right)}}}\cdot{\left({x}\cdot{\sin{{\left({x}+{h}\right)}}}-{\left({x}+{h}\right)}\cdot{\sin{{\left({x}\right)}}}\right)}.$

The denominator tends to $\displaystyle{{x}}^{{2}}$ , the numerator is equal to

$\displaystyle{x}\cdot{\sin{{\left({x}\right)}}}{\cos{{\left({h}\right)}}}+{x}\cdot{\cos{{\left({x}\right)}}}{\sin{{\left({h}\right)}}}-{x}\cdot{\sin{{\left({x}\right)}}}-{h}\cdot{\sin{{\left({x}\right)}}}=$

$\displaystyle={x}\cdot{\sin{{\left({x}\right)}}}{\left({\cos{{\left({h}\right)}}}-{1}\right)}+{x}\cdot{\cos{{\left({x}\right)}}}{\sin{{\left({h}\right)}}}-{h}\cdot{\sin{{\left({x}\right)}}}.$

Dividing this by h as required and using the given limits we obtain for $\displaystyle{h}-\gt{0}$

$\displaystyle{x}\cdot{\sin{{\left({x}\right)}}}\frac{{{\cos{{\left({h}\right)}}}-{1}}}{{h}}+{x}\cdot{\cos{{\left({x}\right)}}}\frac{{\sin{{\left({h}\right)}}}}{{h}}-{\sin{{\left({x}\right)}}}-\gt{0}+{x}\cdot{\cos{{\left({x}\right)}}}-{\sin{{\left({x}\right)}}}.$

Recall the denominator $\displaystyle{{x}}^{{2}}$ and the derivative is

$\displaystyle\frac{{{x}\cdot{\cos{{\left({x}\right)}}}-{\sin{{\left({x}\right)}}}}}{{{x}}^{{2}}},$ which is correct.

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Friday, March 21, 2014

Differentiate $\displaystyle{f{{\left({x}\right)}}}={\left(\frac{{1}}{{x}}\right)}{\sin{{x}}}$ by using the definition of the derivative. Hints: You may use any of the following: $\displaystyle{\sin{{\left({x}+{h}\right)}}}={\sin{{x}}}{\cos{{h}}}+\ldots$

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Thomas Jefferson's election in 1800 is sometimes called the Revolution of 1800. Why could it be described in this way?

Friday, March 21, 2014

Differentiate by using the definition of the derivative. Hints: You may use any of the following:

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Thomas Jefferson&#39;s election in 1800 is sometimes called the Revolution of 1800. Why could it be described in this way?

Differentiate $\displaystyle{f{{\left({x}\right)}}}={\left(\frac{{1}}{{x}}\right)}{\sin{{x}}}$ by using the definition of the derivative. Hints: You may use any of the following: $\displaystyle{\sin{{\left({x}+{h}\right)}}}={\sin{{x}}}{\cos{{h}}}+\ldots$

Thomas Jefferson's election in 1800 is sometimes called the Revolution of 1800. Why could it be described in this way?