frac{d}{dx}(5^{log x}) = dots | vedclass

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Question

(A) Let $y = 5^{\log x}$.Using the property $a^{\log_b c} = c^{\log_b a}$, we can write $y = x^{\log 5}$.Now, differentiate with respect to $x$ using

Vedclass · Accepted Answer

$\frac{5^{\log x} \ln 5}{x}$

Vedclass · Answer

(A) Let $y = 5^{\log x}$.Using the property $a^{\log_b c} = c^{\log_b a}$, we can write $y = x^{\log 5}$.Now, differentiate with respect to $x$ using the power rule $\frac{d}{dx}(x^n) = nx^{n-1}$:$\frac{dy}{dx} = (\log 5) x^{\log 5 - 1}$.Alternatively, using the chain rule for $y = 5^{\log x}$:$\frac{dy}{dx} = 5^{\log x} \cdot \ln 5 \cdot \frac{d}{dx}(\log x) = 5^{\log x} \cdot \ln 5 \cdot \frac{1}{x} = \frac{5^{\log x} \ln 5}{x}$.Since $\ln 5$ is equivalent to $\log_e 5$, the correct expression is $\frac{5^{\log x} \ln 5}{x}$.

$\frac{d}{dx}(5^{\log x}) = \dots$

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