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  • Explain solution RD Sharma class 12 Chapter 11 Higher Order Derivatives exercise Fill in the blanks question 4

Explain solution RD Sharma class 12 Chapter 11 Higher Order Derivatives exercise Fill in the blanks question 4

Answers (1)

Answer:  \frac{d^{2} y}{d x^{2}}=e^{-x}

Hint: Differentiate y w.r.t x twice to get  \frac{d^{2} y}{d x^{2}}

Given:  y=1-x+\frac{x^{2}}{2 !}-\frac{x^{3}}{3 !}+\frac{x^{4}}{4 !} \ldots \ldots

Solution:

It is given that:  y=1-x+\frac{x^{2}}{2 !}-\frac{x^{3}}{3 !}+\frac{x^{4}}{4 !} \ldots \ldots

\therefore \frac{d y}{d x}=-1+\frac{2 x}{2 !}-\frac{3 x^{2}}{3 !}+\frac{4 x^{3}}{4 !}-\ldots

Again, differentiating w.r.t x:

\begin{aligned} &\therefore \frac{d^{2} y}{d x^{2}}=\frac{2}{2 !}-\frac{3(2) x}{3 !}+\frac{4(3) x^{2}}{4 !}-\ldots \\ & \end{aligned}

=\frac{2}{2}-\frac{6 x}{6}+\frac{12 x^{2}}{24}-\ldots \\

=1-x+\frac{x^{2}}{2 !}-\ldots .

We know that:

e^{-x}=1-x+\frac{x^{2}}{2 !}-\frac{x^{3}}{3 !}+\ldots

Thus,  \frac{d^{2} y}{d x^{2}}=e^{-x}.

Posted by

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