Partial Derivative/Examples/x sine y z

Example of Partial Derivative

Let $\map f {x, y, z} = x \map \sin {y z}$ be a real function of $3$ variables.

Then:

$\map {f_3} {a, 1, \pi} = -a$

Proof

By definition, the partial derivative with respect to the $3$rd variable $z$ is obtained by holding $1$st and $2$nd ones constant.

\(\ds \map {f_3} {x, y, z}\)

\(=\)

\(\ds \map {\dfrac \partial {\partial z} } {x \map \sin {y z} }\)

\(\ds \)

\(=\)

\(\ds x y \map \cos {y z}\)

Derivative of $\sin a x$

\(\ds \leadsto \ \ \)

\(\ds \map {f_3} {a, 1, \pi}\)

\(=\)

\(\ds a \times 1 \times \cos \pi\)

\(\ds \)

\(=\)

\(\ds a \times \paren {-1}\)

Cosine of Multiple of Pi

\(\ds \)

\(=\)

\(\ds -a\)

$\blacksquare$

Sources

• 1961: David V. Widder: Advanced Calculus (2nd ed.) ... (previous) ... (next): $1$ Partial Differentiation: $\S 1$. Introduction: $1.1$ Partial Derivatives: Example $\text B$