Question

compute the derivative. use logarithmic differentiation where appropriate
\\( \frac { d } { d x } x ^ { - 14 x } \\)
\\( \frac { d } { d x } x ^ { - 14 x } = \square \\)

Explanation:

Step1: Let \( y = x^{-14x} \)

Take the natural logarithm of both sides: \( \ln y=\ln(x^{-14x})=-14x\ln x \)

Step2: Differentiate both sides with respect to \( x \)

Using the product rule \((uv)^\prime = u^\prime v+uv^\prime\) where \( u = - 14x\), \(u^\prime=-14\) and \(v=\ln x\), \(v^\prime=\frac{1}{x}\)
\(\frac{1}{y}y^\prime=-14\ln x-14x\times\frac{1}{x}=-14\ln x - 14\)

Step3: Solve for \( y^\prime \)

Multiply both sides by \( y=x^{-14x}\)
\(y^\prime=x^{-14x}(-14\ln x - 14)=-14x^{-14x}(1 + \ln x)\)

Answer:

\(-14x^{-14x}(1+\ln x)\)