Question

Find the derivative of the function. \(y=x\left(6^{-2 x}\right)\)

Short answer

The derivative of the function \(y=x*6^{-2x}\) is \(y' = 6^{-2x} - 2x*6^{-2x}*ln(6)\).

Step-by-step solution

Identification of functions

Identify the first function as \(u=x\) and the second function as \(v=6^{-2x}\).

Compute the Derivatives

Calculate the derivatives of \(u\) and \(v\). The derivative of \(u\) with respect to \(x\) is \(u' = 1\). For the derivative of \(v\) with respect to \(x\), use the power rule of exponentials which results in \(v'=-2*6^{-2x}*ln(6)\).

Apply the Product rule

Apply the product rule which states \( (uv)' = u'v + uv'\). Substitute \(u\), \(v\), \(u'\), and \(v'\) into the equation. This leads to the derivative as \(y' = 1* 6^{-2x} - x*2*6^{-2x}*ln(6)\).