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  • Differentiate each of the functions with respect to ‘x’ Differentiate using first principle cos (x^2 + 1)

Differentiate each of the functions with respect to ‘x’

Differentiate using first principle \cos (x^{2} + 1)

Answers (1)

\\ \text{Let f} \left( x \right) =\cos \left( x^{2}+1 \right) \ldots \ldots .. \left( i \right) \\ \\ f \left( x+ \Delta x \right) =\cos \left( \left( x+ \Delta x \right) ^{2}+1 \right) \ldots \ldots \left( ii \right) \\ \\ \text{~ Subtracting equation } \left( i \right) from \: \: equation \left( ii \right) \\ \\ \frac{f \left( x+ \Delta x \right) -f \left( x \right) }{ \Delta x}=\frac{\cos \left( \left( x+ \Delta x \right) ^{2}+1 \right) -\cos \left( x^{2}+1 \right) }{ \Delta x} \\ \\
  \\ =\mathop{\lim }_{ \Delta x \rightarrow 0}\frac{\cos \left( \left( x+ \Delta x \right) ^{2}+1 \right) -\cos \left( x^{2}+1 \right) }{ \Delta x} \\ \\ =\mathop{\lim }_{ \Delta x \rightarrow 0}\frac{-2\sin \left[ \frac{ \left( x+ \Delta x \right) ^{2}+1-x^{2}-1}{2} \right] \sin \left[ \frac{ \left( x+ \Delta x \right) ^{2}+1+x^{2}+1}{2} \right] }{ \Delta x} \\ \\ =\mathop{\lim }_{ \Delta x \rightarrow 0}\frac{-2\sin \left[ \frac{2x \Delta x+ \left( \Delta x \right) ^{2}}{2} \right] \sin \left[ \frac{2x^{2}+2x \Delta x+2}{2} \right] }{ \Delta x} \\ \\ =\mathop{\lim }_{ \Delta x \rightarrow 0} \left( \frac{2x+ \Delta x}{2} \right) \left( ~\frac{-2\sin \left[ \frac{2x \Delta x+ \left( \Delta x \right) ^{2}}{2} \right] \sin \left[ \frac{2x^{2}+2x \Delta x+2}{2} \right] }{\frac{ \Delta x \left( 2x+ \Delta x \right) }{2}} \right) \\ \\
   \\ =\mathop{\lim }_{ \Delta x \rightarrow 0} \left( \frac{2x+ \Delta x}{2} \right) \left( -2\sin \left[ \frac{2x^{2}+2x \Delta x+2}{2} \right] \right) \left( ~\frac{\sin \left[ \frac{2x \Delta x+ \left( \Delta x \right) ^{2}}{2} \right] }{\frac{ \Delta x \left( 2x+ \Delta x \right) }{2}} \right) \\ \\ =- \left( \frac{2x+0}{2} \right) \left( 2\sin \left( x^{2}+0+1 \right) \right) \left( 1 \right) =-2x\sin \left( x^{2}+1 \right) \text{is the required answer} \\ \\

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