Question

Use logarithmic differentiation to find the derivative of y with respect to xy = (10x + 2)^x

Answer 1

Given: An equation-

`y=(10x+2)^x`

Required: To determine the differentiation of y with respect to x.

Explanation: The differentiation of a logarithmic function is-

`\begin{gathered} y=a^x \\ (dy)/(dx)=a^x\ln(a) \end{gathered}`

Taking log both sides on the given equation as-

`\begin{gathered} \ln y=\ln(10x+2)^x \\ =x\ln(10x+2) \end{gathered}`

Now, differentiating with respect to x using product rule as-

`(1)/(y)(dy)/(dx)=\ln(10x+2)(d)/(dx)(x)+x(d)/(dx)\ln(10x+2)`

Further simplifying as-

`(dy)/(dx)=y[\ln(10x+2)+(10x)/(10x+2)]`

Substituting the value of y as-

`(dy)/(dx)=(10x+2)^x[\ln(10x+2)+(10x)/(10x+2)]`

Final Answer: Option D is correct.