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Derivatives of logarithmic functions are mainly based on the chain rule. However, we can generalize it for any differentiable function with a logarithmic function. The differentiation of log is only under the base \(e,\) but we can differentiate under other bases, too.
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For each of the following, differentiate the function first using any rule you want, then using logarithmic differentiation: 1. y = x2 Solution. If y = x2, then lny = ln(x2) = 2lnx. Differentiating, 1 y dy dx = 2 x, so dy dx = 2y x = 2x2 x = 2x. This is the same answer that we could have gotten with the power rule. 2. y = ex Solution. y ...
22 kwi 2024 · For exercises 16 - 23, use logarithmic differentiation to find \(\dfrac{dy}{dx}\). 16) \(y=x^{\sqrt{x}}\) 17) \(y=(\sin 2x)^{4x}\) Answer: \(\dfrac{dy}{dx} = (\sin 2x)^{4x}[4⋅\ln(\sin 2x)+8x⋅\cot 2x]\) 18) \(y=(\ln x)^{\ln x}\) 19) \(y=x^{\log_2 x}\) Answer: \(\dfrac{dy}{dx} = x^{\log_2 x}⋅\dfrac{2\ln x}{x\ln 2}\) 20) \(y=(x^2−1)^{\ln x}\)
17 sie 2024 · Use logarithmic differentiation to determine the derivative of a function. So far, we have learned how to differentiate a variety of functions, including trigonometric, inverse, and implicit functions. In this section, we explore derivatives of exponential and logarithmic functions.
24 maj 2024 · How to find the derivatives of natural and common logarithmic functions with rules, formula, proof, and examples.
Derivative of the Logarithmic Function. Now that we have the derivative of the natural exponential function, we can use implicit differentiation to find the derivative of its inverse, the natural logarithmic function.
We use the rules for the differentiation of logarithmic functions in conjunction with the standard rules for differentiation, that is, the product, quotient, and chain rules.
