compute the derivative. use logarithmic differentiation where appropriate\n\\( \\frac { d } { d x } x ^ {…

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

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

Answer

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))