Textbook Question
9–61. Evaluate and simplify y'.
y = 2x² cos^−1 x+ sin^−1 x
56
views
6. Derivatives of Inverse, Exponential, & Logarithmic Functions
Derivatives of Inverse Trigonometric Functions
Struggling with Calculus?
Join thousands of students who trust us to help them ace their exams!Watch the first videoMultiple Choice
Which of the following is the correct derivative of with respect to ?
A
B
C
D
Verified step by step guidance1
Step 1: Recall the derivative formula for the arcsine function. The derivative of y = arcsin(x) with respect to x is given by \( \frac{1}{\sqrt{1 - x^2}} \). This formula is derived from the chain rule and the inverse trigonometric function properties.
Step 2: Understand the domain of the arcsine function. The arcsine function is defined for \( -1 \leq x \leq 1 \), and the derivative formula \( \frac{1}{\sqrt{1 - x^2}} \) is valid within this interval.
Step 3: Compare the given options. The correct derivative formula matches \( \frac{1}{\sqrt{1 - x^2}} \), which is one of the provided choices.
Step 4: Verify the formula by differentiating y = arcsin(x) using implicit differentiation. Let y = arcsin(x), then \( \sin(y) = x \). Differentiating both sides with respect to x gives \( \cos(y) \frac{dy}{dx} = 1 \), and solving for \( \frac{dy}{dx} \) yields \( \frac{dy}{dx} = \frac{1}{\cos(y)} \). Using the Pythagorean identity \( \cos^2(y) = 1 - \sin^2(y) \), substitute \( \sin(y) = x \) to get \( \cos(y) = \sqrt{1 - x^2} \). Thus, \( \frac{dy}{dx} = \frac{1}{\sqrt{1 - x^2}} \).
Step 5: Conclude that the correct derivative of y = arcsin(x) with respect to x is \( \frac{1}{\sqrt{1 - x^2}} \), as derived and verified.
7:26mWatch next
Master Derivatives of Inverse Sine & Inverse Cosine with a bite sized video explanation from Patrick
Start learningRelated Videos
Related Practice
Derivatives of Inverse Trigonometric Functions practice set
Problem sets built by lead tutorsExpert video explanations