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4th Edition

James Stewart

Publisher: Cengage Learning

ISBN: 9781337687805

Chapter 3.5, Problem 2E

To determine

**To calculate: **The first derivative of the function

Expert Solution

The first derivative of the function is

**Given information:**

The function

**Formula used:**

Thechain rule for differentiation is if f is a function of gthen

Power rule for differentiation is

**Calculation:**

Consider the function

Differentiate both sides with respect to x ,

Recall that power rule for differentiation is

Apply it. Also observe that y is a function of x,

Isolate the value of

Thus, the first derivative of the function is

To determine

**To calculate: **The first derivative of the function

Expert Solution

The first derivative of the function is

**Given information:**

The function

**Formula used:**

Thechain rule for differentiation is if f is a function of gthen

Power rule for differentiation is

Quotient rule for differentiation is

Product rule for differentiation is

**Calculation:**

Consider the function

Isolate the value of y on left hand side on the above equation.

Subtract the terms

Square both the sides of the above expression,

Differentiate both sides with respect to x ,

Recall that power rule for differentiation is

Apply it. Also observe that y is a function of x,

Thus, the first derivative of the function is

To determine

**To verify: **The solutions obtained in parts

Expert Solution

**Given information:**

The derivative of the function

Consider the derivative of the function

In order to verify that derivative obtained implicitly and explicitly is same substitute the value of y from the function in the value of derivative found by implicit differentiation.

Consider the function

Isolate the value of y on left hand side on the above equation.

Subtract the terms

The first derivative of the function by implicit differentiation is

Now,

Since, the derivative found implicitly is equal to derivative found explicitly so both the derivatives are equal.

Hence, the solutions obtained in parts