What is the derivative of x cos ⁡ ( π x ) ?

Question

What is the derivative of             x              cos      ⁡      (      π      x      )      ?

Aliana Porter · 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) = \sqrt{x}

and

g (x) = \cos (π x)

\frac{\cos (π x) \frac{d}{d x} [\sqrt{x}] - \sqrt{x} \frac{d}{d x} [\cos (π x)]}{\cos^{2} (π x)}

Differentiate using the Power Rule.

\frac{\cos (π x) (\frac{1}{2} x^{\frac{1}{2} - 1}) - \sqrt{x} \frac{d}{d x} [\cos (π x)]}{\cos^{2} (π x)}

To write -1 as a fraction with a common denominator, multiply by

\frac{2}{2}

\frac{\cos (π x) (\frac{1}{2} x^{\frac{1}{2} - 1 * \frac{2}{2}}) - \sqrt{x} \frac{d}{d x} [\cos (π x)]}{\cos^{2} (π x)}

Combine -1 and

\frac{2}{2}

\frac{\cos (π x) (\frac{1}{2} x^{\frac{1}{2} + \frac{- 1 * 2}{2}}) - \sqrt{x} \frac{d}{d x} [\cos (π x)]}{\cos^{2} (π x)}

Combine the numerators over the common denominator.

\frac{\cos (π x) (\frac{1}{2} x^{\frac{- 1}{2}}) - \sqrt{x} \frac{d}{d x} [\cos (π x)]}{\cos^{2} (π x)}

Simplify.

\frac{\cos (π x) \frac{1}{2 x^{\frac{1}{2}}} - \sqrt{x} \frac{d}{d x} [\cos (π x)]}{\cos^{2} (π 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) = \cos (x)

and

g (x) = π x

.

\frac{\cos (π x) \frac{1}{2 x^{\frac{1}{2}}} - \sqrt{x} (- \sin (π x) \frac{d}{d x} [π x])}{\cos^{2} (π x)}

Differentiate.

\frac{\cos (π x) \frac{1}{2 x^{\frac{1}{2}}} - \sqrt{x} (- π \sin (π x))}{\cos^{2} (π x)}

Simplify.

\frac{2 π x \sin (π x) + \cos (π x)}{2 x^{\frac{1}{2}} \cos^{2} (π x)}

What is the derivative of <msqrt> x </msqrt> cos ⁡<!-- ⁡ -