Find the second order derivatives of the function
\(tan^{-1}x\)
\(tan^{-1}x\)
Solution and Explanation
Let \(y=tan^{-1}x\)
Then,
\(\frac{dy}{dx}=\frac{d}{dx}(tan^{-1}x)=\frac{1}{1+x^2}\)
\(∴\frac{d^2y}{dx^2}=\frac{d}{dx}[\frac{1}{1+x^2}]\)
\(=\frac{d}{dx}(1+x^2)^{-1}\)
\(=(-1).(1+x^2)^{-2}.\frac{d}{dx}(1+x^2)\)
\(=\frac{-1}{(1+x^2)^2}\times 2x\)
\(=\frac{-2x}{(1+x^2)^2}\)
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Alexia Limited invited applications for issuing 1,00,000 equity shares of ₹ 10 each at premium of ₹ 10 per share.
The amount was payable as follows:- On application ₹ 9 per share (Including premium ₹ 6 per share)
- On allotment ₹ 8 per share (Including premium ₹ 4 per share)
- On first and final call ₹ 3 per share.
Applications were received for 1,50,000 equity shares and allotment was made to the applicants as follows:
Category A: Applicants for 90,000 shares were allotted 70,000 shares.
Category B: Applicants for 60,000 shares were allotted 30,000 shares.
Excess money received on application was adjusted towards allotment and first and final call.
Shekhar, who had applied for 1200 shares failed to pay the first and final call. Shekhar belonged to category B.
Pass necessary journal entries for the above transactions in the books of Alexia Limited. Open calls in arrears and calls in advance account, wherever necessary.- The firm of Amish, Nitish and Misha, who have been sharing profits in the ratio of 2 : 2 : 1, have existed for some years. Misha wanted that she should get equal share in the profits with Amish and Nitish and she further wished that the change in the profit sharing ratio should come into effect retrospectively for the last three years. Amish and Nitish had agreement for this.
The profits for the last three years were: 2021–22 ₹ 1,15,000; 2022–23 ₹ 1,24,000; 2023–24 ₹ 2,11,000
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Concepts Used:
Second-Order Derivative
The Second-Order Derivative is the derivative of the first-order derivative of the stated (given) function. For instance, acceleration is the second-order derivative of the distance covered with regard to time and tells us the rate of change of velocity.
As well as the first-order derivative tells us about the slope of the tangent line to the graph of the given function, the second-order derivative explains the shape of the graph and its concavity.
The second-order derivative is shown using \(f’’(x)\text{ or }\frac{d^2y}{dx^2}\).