Partial Derivatives Examples 2

# Partial Derivatives Examples 2

We will now look at even more examples of computing partial derivatives for functions of several variables. Be sure to review the Partial Derivatives page beforehand! More examples can be found on the Partial Derivatives Examples 1 page.

Before we look at the following examples, it will be important to recall the following differentiation rules for single variable real-valued functions:

• Rule for Power Functions: $\frac{d}{dx} x^n = nx^{n-1}$.
• Rules for Trigonometric Functions: $\frac{d}{dx} \sin x = \cos x$, $\frac{d}{dx} \cos x = - \sin x$, and $\frac{d}{Dx} \tan x = \sec ^2 x$.
• Rules for Exponential Functions: $\frac{d}{dx} a^x = a^x \ln (a)$ and $\frac{d}{dx} e^x = e^x$.
• Rules for Logarithmic Functions: $\frac{d}{dx} \log_a (x) = \frac{1}{x \ln a}$ and $\frac{d}{dx} \ln x = \frac{1}{x}$.
• The Chain Rule: $\frac{d}{dx} f(g(x)) = f'(g(x)) g'(x)$.
• The Product Rule: $\frac{d}{dx} (f(x)g(x)) = f'(x)g(x) + f(x)g'(x)$.
• The Quotient Rule: $\frac{d}{dx} \left ( \frac{f(x)}{g(x)} \right ) = \frac{g(x)f'(x) - f(x)g'(x)}{[g(x)]^2}$ provided that $g(x) \neq 0$.

## Example 1

Let $z = \ln (x^5e^x 2^y \cos y )$. Find $\frac{\partial z}{\partial x}$ and $\frac{\partial z}{\partial y}$.

To compute $\frac{\partial z}{\partial x}$ we will use the rule for logarithmic differentiation and the product rule:

(1)
\begin{align} \quad \frac{\partial z}{\partial x} = \frac{(5x^4e^x + x^5e^x)(2^y \cos y)}{x^5e^x 2^y \cos y} = \frac{5x^4e^x 2^y \cos y + x^5e^x 2^y \cos y}{x^5e^x 2^y \cos y} = \frac{5}{x} + 1 \end{align}

To compute $\frac{\partial z}{\partial y}$ we will also use the rule for logarithmic differentiation and the product rule:

(2)
\begin{align} \quad \frac{\partial z}{\partial y} = \frac{(x^5e^x)(2^y \ln 2 \cos y - 2^y \sin y)}{x^5e^x 2^y \cos y} = \frac{x^5e^x2^y\ln 2 \cos y - x^5e^x 2^y \sin y}{x^5 e^x 2^y \cos y} = \ln 2 - \frac{\sin y}{\cos y} = \ln 2 - \tan y \end{align}

## Example 2

Let $w = 2 \sin (xy) \cos (yz) \tan (xz)$. Find $\frac{\partial w}{\partial x}$, $\frac{\partial w}{\partial y}$, and $\frac{\partial w}{\partial z}$.

To compute $\frac{\partial x}{\partial w}$ we will need to use the product rule:

(3)
\begin{align} \frac{\partial w}{\partial x} = 2 \cos (yz) [ y \cos (xy)\tan (xz) + z\sin (yz) \sec ^2 (xz) ] \end{align}

To compute $\frac{\partial y}{\partial w}$ we will also need to use the product rule:

(4)
\begin{align} \frac{\partial w}{\partial y} = 2 \tan (xz) [x \cos (xy) \cos (yz) - z\sin (xy) \sin (yz)] \end{align}

And lastly, to compute $\frac{\partial w}{\partial z}$ we will once again need to use the product rule:

(5)
\begin{align} \frac{\partial w}{\partial z} = 2\sin (xy) [-y \sin (yz) \tan (xz) + x\cos (yz) \sec ^2 (xz) ] \end{align}

## Example 3

Let $w = \cos ( \sin (x^2y^4z^5)) + \ln (xyz)$. Find $\frac{\partial w}{\partial x}$, $\frac{\partial w}{\partial y}$, and $\frac{\partial w}{\partial z}$.

Applying the chain rule we obtain that:

(6)
\begin{align} \quad \frac{\partial w}{\partial x} = -\sin (\sin (x^2y^4z^5)) \cdot \cos (x^2y^4z^5) \cdot [2xy^4z^5] + \frac{1}{z} \end{align}
(7)
\begin{align} \quad \frac{\partial w}{\partial y} = -\sin ( \sin (x^2y^4z^5)) \cdot \cos (x^2 y^4 z^5) \cdot [4x^2y^3z^5] + \frac{1}{y} \end{align}
(8)
\begin{align} \quad \frac{\partial w}{\partial z} = -\sin (\sin (x^2y^4z^5)) \cdot \cos (x^2 y^4 z^5) \cdot [5x^2 y^4 z^4] + \frac{1}{z} \end{align}

## Example 4

Let $w = \frac{x^2y + y\cos z}{e^x + y^2e^z}$. Find $\frac{\partial w}{\partial x}$, $\frac{\partial w}{\partial y}$, and $\frac{\partial w}{\partial z}$.

Applying the quotient rule and we get that:

(9)
\begin{align} \quad \frac{\partial w}{\partial x} = \frac{(e^x + y^2e^z)(2xy) - (x^2y + y\cos z)(e^x)}{(e^x + y^2e^z)^2} \end{align}
(10)
\begin{align} \quad \frac{\partial w}{\partial y} = \frac{(e^x + y^2e^z)(x^2 -y \sin z) - (x^2y + y\cos z)(2ye^z)}{(e^x + y^2e^z)^2} \end{align}
(11)
\begin{align} \quad \frac{\partial w}{\partial z} = \frac{(e^x + y^2e^z)(-y \sin z) - (x^2y + y\cos z)(y^2e^z)}{(e^x + y^2e^z)^2} \end{align}