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Theorems for Differentiability

Y = f( 1 /x) ...

Question

Y = f( 1 /x) and f'(x) = sin( x ^2) then dy/dx is

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y = f (sin x)

f^{'} (x) = {sin}^{- 1} x

\frac{d y}{d x}

f^{'} (x) = sin x^{2}

y = f (x^{2} + 1)

\frac{d y}{d x}

x = 2

x = f^{''} (t) c o s t + f^{'} (t) s i n t

y = {- f}^{''} (t) s i n t + f^{'} (t) c o s t .

\int {[{(\frac{d x}{d t})}^{2} + {(\frac{d y}{d t})}^{2}]}^{\frac{1}{2}} d t

f (x), f^{'} (x), f^{''} (x), f^{'''} (x) > 0

f^{^{'}} (x) = \sqrt{2 x^{2} - 1}

y = f (x^{2})

\frac{d y}{d x}

f^{'} (x) = \sqrt{2 x^{2} - 1}

y = f^{'} (x^{2})

\frac{d y}{d x}

x = 1

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