Share
Calculus Chain Rule and Implicit Differentiation Problems with Solutions
Published on July 15, 2026
Calculus
This resource provides step-by-step solutions for calculus problems involving the chain rule, implicit differentiation, and finding tangent and normal lines to curves.
Question
Answer
This assignment contains several calculus problems involving the chain rule, implicit differentiation, and finding tangent/normal lines. Here are the step-by-step solutions for each question.
Question 1: Find the derivatives using the chain rule.
a.
First, rewrite the function using fractional exponents:
Let
Using the chain rule
First, rewrite the function using fractional exponents:
Let
Using the chain rule
b.
Let
Using the chain rule
Let
Using the chain rule
c.
Rewrite the function:
Let
Rewrite the function:
Let
Question 2: Find the derivatives using the chain rule.
a.
Rewrite as
Let
(Note: Using double angle identity, this can also be written as
Rewrite as
Let
(Note: Using double angle identity, this can also be written as
b.
Let
Let
c.
Let
Let
Question 3: Consider the curve
a. Calculate the coordinates of the two points on the curve for which
Substitute
The two points are
Substitute
The two points are
b. Show that at one of the two points, the tangent is parallel to the x-axis.
A tangent is parallel to the x-axis if the slope
Use implicit differentiation on
A tangent is parallel to the x-axis if the slope
Use implicit differentiation on
Check point
Since the slope is
Since the slope is
c. Determine the equations of the tangent and normal lines at the second point
At
At
-
Tangent Line:
Using point-slope form:
-
Normal Line:
The slope of the normal is the negative reciprocal:.