Differentiate the following w.r.t. \(x\):
\(\frac{cos\,x}{log\,x},x>0\)
\(\frac{cos\,x}{log\,x},x>0\)
Solution and Explanation
Let \(y=\frac{cos\,x}{log\,x}\)
By using the chain rule,we obtain
\(\frac{dy}{dx}=\frac{\frac{d}{dx}(cos\,x)\times log\,x-cos\,x\times \frac{d}{dx}(log\,x)}{(log\,x)^2}\)
\(=\frac{-sin\,xlog\,x-cos\,x\times \frac{1}{x}}{(log\,x)^2}\)
\(=\frac{-[xlog\,x.sin\,x+cos\,x]}{x(log\,x)^2},x>0\)
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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)