Find the derivative of e x cos ⁡ ( π x ).

Question

Find the derivative of       e          x            ﻿    cos  ⁡  (  π  x  ).

nelppeazy9v3ie · Accepted Answer

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = e^{x}$ and $g (x) = \cos (π x)$ .

$e^{x} \frac{d}{d x} [\cos (π x)] + \cos (π x) \frac{d}{d x} [e^{x}]$

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 $g (x) = π x$ .

$e^{x} (- \sin (π x) \frac{d}{d x} [π x]) + \cos (π x) \frac{d}{d x} [e^{x}]$

Differentiate.

$e^{x} (- \sin (π x) π) + \cos (π x) \frac{d}{d x} [e^{x}]$

Differentiate using the Exponential Rule which states that $\frac{d}{d x} [a^{x}]$ is $a^{x} \ln (a)$ where $a$ = $e$ .

$e^{x} (- \sin (π x) π) + \cos (π x) e^{x}$

Reorder terms.

$- π e^{x} \sin (π x) + e^{x} \cos (π x)$

Find the derivative of e <mrow class="MJX-TeXAtom-ORD"> x </mrow> </msup>

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