What is the derivative of tan ⁡ ( π x ) ln ⁡ ( x ) ?

Question

What is the derivative of             tan      ⁡      (      π      x      )              ln      ⁡      (      x      )      ?

Leia Wiggins · Accepted Answer

Differentiate using the Quotient Rule which states that

\frac{d}{d x} [\frac{f (x)}{g (x)}]

is

\frac{g (x) \frac{d}{d x} [f (x)] - f (x) \frac{d}{d x} [g (x)]}{g (x)^{2}}

where

f (x) = \tan (π x)

and

g (x) = \ln (x)

\frac{\ln (x) \frac{d}{d x} [\tan (π x)] - \tan (π x) \frac{d}{d x} [\ln (x)]}{\ln^{2} (x)}

Differentiate using the chain rule, which states that

\frac{d}{d x} [f (g (x))]

is f'(g(x))g'(x) where

f (x) = \tan (x)

and

g (x) = π x

\frac{\ln (x) (\sec^{2} (π x) \frac{d}{d x} [π x]) - \tan (π x) \frac{d}{d x} [\ln (x)]}{\ln^{2} (x)}

Differentiate.

\frac{\ln (x) (π \sec^{2} (π x)) - \tan (π x) \frac{d}{d x} [\ln (x)]}{\ln^{2} (x)}

The derivative of

\ln (x)

with respect to x is

\frac{1}{x}

\frac{\ln (x) (π \sec^{2} (π x)) - \tan (π x) \frac{1}{x}}{\ln^{2} (x)}

Simplify.

\frac{π \sec^{2} (π x) \ln (x) - \frac{\tan (π x)}{x}}{\ln^{2} (x)}

What is the derivative of tan ⁡ ( π<!-- π -