Resolved: "Find the first partial derivatives of the following..."

Amari Flowers

Answered question

2021-03-09

Find the first partial derivatives of the following functions.

s (y, z) = z^{2} \tan y z

pattererX

Skilled2021-03-10Added 95 answers

Step 1
Consider the provided function,

S (y, z) = z^{2} \tan y z

Find the first partial derivatives.
So, we find the expression for

\frac{\partial S}{\partial y} and \frac{\partial S}{\partial z}

.
First we find the expression for

\frac{\partial S}{\partial y}

\frac{\partial S}{\partial y} = \frac{\partial}{\partial y} (z^{2} \tan y z)

Step 2
Simplifying further,

\frac{\partial S}{\partial y} = \frac{\partial}{\partial y} (z^{2} \tan y z)

= z^{2} \frac{\partial}{\partial y} (\tan y z)

= z^{2} \sec^{2} y z \frac{\partial}{\partial y} (y z)

= z^{2} \sec^{2} y z \cdot z

= z^{3} \sec^{2} y z

Now, find the expression for

\frac{\partial S}{\partial z}

\frac{\partial S}{\partial z} = \frac{\partial}{\partial z} (z^{2} \tan y z)

Step 3
Simplifying further,

\frac{\partial S}{\partial z} = \frac{\partial}{\partial z} (z^{2}) \tan (y z) + \frac{\partial}{\partial z} (\tan (y z)) z^{2}

= 2 z \cdot \tan (y z) + z^{2} \cdot \sec^{2} (y z) y

= 2 z \tan (y z) + \sec^{2} (y z) y z^{2}

Hence.

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