Question

Math

Posted about 2 months ago

```
Find the differential of the function.
$z=e^{-2 x} \cos 2 \pi t$
```

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Answer from Sia

Posted about 2 months ago

Solution by Steps

step 1

We need to find the differential of the function $z = e^{-2x} \cos(2\pi t)$

step 2

To find the differential, we use the product rule for differentiation. The product rule states that if $u$ and $v$ are functions of $x$ and $t$, then $d(uv) = u \, dv + v \, du$. Here, $u = e^{-2x}$ and $v = \cos(2\pi t)$

step 3

First, we differentiate $u = e^{-2x}$ with respect to $x$:
$\frac{du}{dx} = \frac{d}{dx}(e^{-2x}) = -2e^{-2x}$

step 4

Next, we differentiate $v = \cos(2\pi t)$ with respect to $t$:
$\frac{dv}{dt} = \frac{d}{dt}(\cos(2\pi t)) = -2\pi \sin(2\pi t)$

step 5

Now, applying the product rule:
$dz = e^{-2x} \, d(\cos(2\pi t)) + \cos(2\pi t) \, d(e^{-2x})$

step 6

Substituting the derivatives we found:
$dz = e^{-2x} (-2\pi \sin(2\pi t) \, dt) + \cos(2\pi t) (-2e^{-2x} \, dx)$

step 7

Simplifying the expression:
$dz = -2\pi e^{-2x} \sin(2\pi t) \, dt - 2e^{-2x} \cos(2\pi t) \, dx$

Answer

The differential of the function $z = e^{-2x} \cos(2\pi t)$ is $dz = -2\pi e^{-2x} \sin(2\pi t) \, dt - 2e^{-2x} \cos(2\pi t) \, dx$.

Key Concept

Differentiation using the product rule

Explanation

The product rule is used to differentiate a product of two functions. In this case, we differentiated $e^{-2x}$ and $\cos(2\pi t)$ separately and then applied the product rule to find the differential of the given function.

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