Chapter 14, Problem 2T

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11th Edition

Ronald J. Harshbarger + 1 other

ISBN: 9781305108042

Textbook Problem

Find all first and second partial derivatives of

To determine

**To calculate:** All the first and second partial derivatives of

Explanation

**Given Information:**

The provided function is

**Formula used:**

For a function *f* with respect to *x* is calculated by taking the derivative of *x* and keeping the other variable *y* constant. The partial derivative of *f* with respect to *x* is denoted by *f* with respect to *y* is denoted by

For a function

(1) When both derivatives are taken with respect to *x* is

(2) When both derivatives are taken with respect to *y* is

(3) When first derivative is taken with respect to *x* and second derivative is taken with respect to *y* is

(4) When first derivative is taken with respect to *y* and second derivative is taken with respect to *x* is

Power of

Constant function rule for a constant

Chain rule for function

Coefficient rule for a constant

**Calculation:**

Consider the function,

Recall that, for a function *f* with respect to *x* is calculated by taking the derivative of *x* and keeping the other variable *y* constant.

Use the power of *x* rule for derivatives, the constant function rule, the chain rule and the coefficient rule,

And,

Hence, the first partial derivatives of the function

Recall that, for a function *x* is

Use the power of *x* rule for derivatives, the constant function rule and the coefficient rule.

Simplify it further,

Recall that, for a function *x* and second derivative is taken with respect to *y* is

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