derivative of sec(x) and csc(x), quotient rule, power + chain rule

TL;DR

This video explains how to find the derivative of trigonometric functions using different methods.

Transcript

it would be differentiating seek an ex and one of the ways to do it is of course you can use the definition and can leave a comment down below and let us know how that goes but I will choose to differentiate one over cosine X instead and if you look at this power here we can just use the quotient rule and I will do that for you guys first which we ... Read More

Key Insights

• 📏 The derivative of sec(x) is sec(x) * tan(x), obtained from the quotient rule.
• ✊ An alternative method for finding the derivative of sec(x) is to rewrite it as (cos(x))^(-1) and apply the power rule.
• 📏 The derivative of cosec(x) can be found using both the quotient rule (-cosec(x) * cot(x)) and the power rule (-cos(x)/(sin(x))^2).

Questions & Answers

Q: How do you differentiate the function f(x) = sec(x) using the quotient rule?

To differentiate f(x) = sec(x), apply the quotient rule. Square the denominator (cos(x)), multiply it by the derivative of the numerator (0), and subtract the product of the numerator (1) and the derivative of the denominator (-sin(x)). The result is sec(x) * tan(x).

Q: How can f(x) = sec(x) be differentiated using the power rule?

Rewrite f(x) = sec(x) as f(x) = (cos(x))^(-1). Apply the power rule by bringing the exponent (-1) down and subtracting 1, giving us -1 * cos(x)^(-2). Multiply by the derivative of cos(x) (-sin(x)) to obtain -sin(x)/(cos(x))^2, which simplifies to -sec^2(x).

Q: What is the derivative of cosec(x) using the quotient rule?

To differentiate cosec(x), use the quotient rule. Square the denominator (sin(x)), multiply it by the derivative of the numerator (1), and subtract the product of the numerator (cosec(x)) and the derivative of the denominator (cos(x)). The result is -cosec(x) * cot(x).

Q: How is the derivative of cosec(x) derived using the power rule?

Rewrite cosec(x) as (sin(x))^(-1). Apply the power rule by bringing the exponent (-1) down and subtracting 1, resulting in -sin(x)^(-2). Multiply by the derivative of sin(x) (cos(x)) to obtain -cos(x)/(sin(x))^2, which simplifies to -cosec^2(x).

Summary & Key Takeaways

• The video demonstrates how to differentiate the function f(x) = sec(x) using the quotient rule.

• An alternative method is shown, where f(x) = sec(x) is rewritten as f(x) = (cos(x))^(-1) to use the power rule.

• The derivative of cosec(x) is derived using both the quotient rule and the power rule.

• A quick overview is given on the derivative of sine(x).