Differentiate the following w.r.t. \(x\):
\(log(cos\,e^x)\)
\(log(cos\,e^x)\)
Solution and Explanation
Let \(y=log(cos\,e^x)\)
By using the chain rule,we obtain
\(\frac{dy}{dx}=\frac{d}{dx}[log(cos\,e^x)]\)
\(=\frac{1}{cos\,e^x}.\frac{d}{dx}(cos\,e^x)\)
\(=\frac{1}{cos\,e^x}.(-sin\,e^x).\frac{d}{dx}(e^x)\)
\(=\frac{-sin\,e^x}{cos\,e^x}.e^x\)
\(=-e^xtan\,e^x,e^x≠(2n+1)\frac{π}{2},n∈N\)
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List-I List-II (A) \( f(x) = |x| \) (I) Not differentiable at \( x = -2 \) only (B) \( f(x) = |x + 2| \) (II) Not differentiable at \( x = 0 \) only (C) \( f(x) = |x^2 - 4| \) (III) Not differentiable at \( x = 2 \) only (D) \( f(x) = |x - 2| \) (IV) Not differentiable at \( x = 2, -2 \) only
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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)