Differentiate the following w.r.t. \(x\):
\(e^x+e^{x^2}+....+e^{x^5}\)
\(e^x+e^{x^2}+....+e^{x^5}\)
Solution and Explanation
Let \(y=e^x+e^{x^2}+....+e^{x^5}\)
\(\frac{d}{dx}=\frac{d}{dx}[e^x+e^{x^2}+....+e^{x^5}]\)
\(=\frac{d}{dx}(e^x)+\frac{d}{dx}(e^{x^2})+\frac{d}{dx}(e^{x^4})+\frac{d}{dx}(e^{x^5})\)
\(=e^x+[e^{x^2}.\frac{d}{dx}(x^2)]+[e^{x^3}.\frac{d}{dx}(x^3)]+[e^{x^4}.\frac{d}{dx}(x^4)]+[e^{x^5}.\frac{d}{dx}(x^5)]\)
\(=e^x+2xe^{x^2}+3x^2e^{x^3}+4x^3e^{x^4}+5x^4e^{x^3}\)
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Questions Asked in CBSE CLASS XII exam
Bittu and Chintu were partners in a firm sharing profit and losses in the ratio of 4 : 3. Their Balance Sheet as at 31st March, 2024 was as follows:
On 1st April, 2024, Diya was admitted in the firm for \( \frac{1}{7} \)th share in the profits on the following terms:- (i) New profit sharing ratio between Bittu, Chintu and Diya was 3 : 3 : 1 .
- (ii) Fixed Assets were found to be overvalued by ₹ 1,40,000.
- (iii) Creditors were paid ₹ 4,20,000 in full settlement.
- (iv) Diya brought proportionate capital and ₹ 5,60,000 as her share of goodwill premium by cheque.
Prepare Revaluation Account and Partners' Capital Accounts.
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Concepts Used:
Exponential and Logarithmic Functions
Logarithmic Functions:
The inverses of exponential functions are the logarithmic functions. The exponential function is y = ax and its inverse is x = ay. The logarithmic function y = logax is derived as the equivalent to the exponential equation x = ay. y = logax only under the following conditions: x = ay, (where, a > 0, and a≠1). In totality, it is called the logarithmic function with base a.
The domain of a logarithmic function is real numbers greater than 0, and the range is real numbers. The graph of y = logax is symmetrical to the graph of y = ax w.r.t. the line y = x. This relationship is true for any of the exponential functions and their inverse.
Exponential Functions:
Exponential functions have the formation as:
f(x)=bx
where,
b = the base
x = the exponent (or power)