Step by step sample solution of parametric derivative

Step by step solution returned by our online calculator is given below:

Parametric derivative calculator

Find the derivative of the parametrically defined function:

$$ \left\{ \begin{aligned} &x(t) = 1 + \sin(t) \\ &y(t) = 2 \cdot t - \cos(t) \end{aligned} \right.$$

$t$ - parametric variable.

Input interpretation

Find the derivative $\frac{\mathrm{d}y}{\mathrm{d}x}$ of parametrically defined function:

$$ \left\{ \begin{aligned} &x(t) = 1 + \sin(t) \\ &y(t) = 2 \cdot t - \cos(t) \end{aligned} \right.$$

Answer

$\left(2+\sin\left(t\right)\right)\cdot\sec\left(t\right)$

Step by step solution

Step 1

Formula for derivative calculation of parametrically defined function has the form:

$\frac{\mathrm{d}y(x)}{\mathrm{d}x}=\frac{\frac{\mathrm{d}y(t)}{\mathrm{d}t}}{\frac{\mathrm{d}x(t)}{\mathrm{d}t}}$

Step 2

First calculate the derivative $\frac{\mathrm{d}y(t)}{\mathrm{d}t}$:

$\frac{\mathrm{d}}{\mathrm{d}t}\left(2\cdot t-\cos(t)\right)$

Step 3

Differentiate the sum term by term:

$\frac{\mathrm{d}}{\mathrm{d}t}\left(2\cdot t\right)+\frac{\mathrm{d}}{\mathrm{d}t}\left(-\cos(t)\right)$

Step 4

Factor out constants:

$2\cdot\frac{\mathrm{d}t}{\mathrm{d}t}+\frac{\mathrm{d}}{\mathrm{d}t}\left(-\cos(t)\right)$

Step 5

The derivative of $t$ by $t$ equals to $1$:

$2+\frac{\mathrm{d}}{\mathrm{d}t}\left(-\cos(t)\right)$

Step 6

Factor out constants:

$2-\frac{\mathrm{d}}{\mathrm{d}t}\left(\cos(t)\right)$

Step 7

By using derivative identities $\left(\frac{\mathrm{d}}{\mathrm{d}t}\left(\cos(t)\right)=-\sin(t)\right)$, we get:

$2+\sin(t)$

Step 8

So, we have:

$\frac{\mathrm{d}y(t)}{\mathrm{d}t}=2+\sin(t)$

Step 9

Now, calculate the derivative $\frac{\mathrm{d}x(t)}{\mathrm{d}t}$:

$\frac{\mathrm{d}}{\mathrm{d}t}(1+\sin(t))$

Step 10

Differentiate the sum term by term:

$\frac{\mathrm{d}1}{\mathrm{d}t}+\frac{\mathrm{d}\sin(t)}{\mathrm{d}t}$

Step 11

Derivative of constant expression equals to zero:

$\frac{\mathrm{d}\sin(t)}{\mathrm{d}t}$

Step 12

By using derivative identities $\left(\frac{\mathrm{d}\sin(t)}{\mathrm{d}t}=\cos(t)\right)$, we get:

$\cos(t)$

Step 13

So, we have:

$\frac{\mathrm{d}x(t)}{\mathrm{d}t}=\cos(t)$

Step 14

Substitute received data into initial formula:

$\frac{\mathrm{d}y(x)}{\mathrm{d}x}=\frac{\frac{\mathrm{d}y(t)}{\mathrm{d}t}}{\frac{\mathrm{d}x(t)}{\mathrm{d}t}}=\frac{2+\sin(t)}{\cos(t)}=\left(2+\sin(t)\right)\cdot \sec(t)$

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