Question
Differentiate the following functions with respect to x:
$10^{\log\sin\text{x}}$
$10^{\log\sin\text{x}}$
✓
Answer
Let $\text{y}=10^{\log\sin\text{x}}\ .....(\text{i})$
Taking log on both sides,
$\log\text{y}=\log10^{\log\sin\text{x}}$
$\Rightarrow\log\text{y}=\log\sin\text{x}\log10$
Differentiating with respect to x,
$\Rightarrow\frac{1}{\text{y}}\frac{\text{dy}}{\text{dx}}=\log10\frac{\text{d}}{\text{dx}}\log\sin\text{x}$
$\Rightarrow\frac{1}{\text{y}}\frac{\text{dy}}{\text{dx}}=\log10\frac{1}{\sin\text{x}}\frac{\text{d}}{\text{dx}}(\sin\text{x})$
$\Rightarrow\frac{1}{\text{y}}\frac{\text{dy}}{\text{dx}}=\log10\Big(\frac{1}{\sin\text{x}}\Big)(\cos\text{x})$
$\Rightarrow\frac{\text{dy}}{\text{dx}}=\text{y}\big[\log10\times\cot\text{x}\big]$
$\Rightarrow\frac{\text{dy}}{\text{dx}}=10^{\log\sin\text{x}}\times\log10\times\cot\text{x}$
[Using equation (i)]
Taking log on both sides,
$\log\text{y}=\log10^{\log\sin\text{x}}$
$\Rightarrow\log\text{y}=\log\sin\text{x}\log10$
Differentiating with respect to x,
$\Rightarrow\frac{1}{\text{y}}\frac{\text{dy}}{\text{dx}}=\log10\frac{\text{d}}{\text{dx}}\log\sin\text{x}$
$\Rightarrow\frac{1}{\text{y}}\frac{\text{dy}}{\text{dx}}=\log10\frac{1}{\sin\text{x}}\frac{\text{d}}{\text{dx}}(\sin\text{x})$
$\Rightarrow\frac{1}{\text{y}}\frac{\text{dy}}{\text{dx}}=\log10\Big(\frac{1}{\sin\text{x}}\Big)(\cos\text{x})$
$\Rightarrow\frac{\text{dy}}{\text{dx}}=\text{y}\big[\log10\times\cot\text{x}\big]$
$\Rightarrow\frac{\text{dy}}{\text{dx}}=10^{\log\sin\text{x}}\times\log10\times\cot\text{x}$
[Using equation (i)]
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