Limit of (sin(a + h) - sin(a))/h as h approaches zero (Two Solutions)

TL;DR

This video explains two methods to evaluate the derivative of sine and concludes that it is equal to cosine.

Transcript

in this problem we have to evaluate the limit as h approaches zero of this expression here so there are two ways to do this so let me show you two different ways so solution one solution one is to recall that the derivative of a function at a is equal to the limit as h approaches 0 of f of a plus h minus f of a over h now normally you use x but i w... Read More

Key Insights

• 😑 There are two methods to evaluate the derivative of sine: using the derivative definition or simplifying the expression.
• 👨‍💼 The derivative of sine is equal to cosine.
• 😑 Trigonometric identities can be used to simplify the expression and evaluate the limit.

Questions & Answers

Q: What is the derivative of sine?

The derivative of sine is cosine. This is proven by evaluating the limit as h approaches zero using either the derivative definition or trigonometric identities.

Q: How can the limit of the expression be evaluated without using the derivative definition?

By applying trigonometric identities to the expression and simplifying it, the limit can be found to be equal to the cosine of a.

Q: What is the significance of the special limits mentioned?

The special limits used in the simplifications (lim h->0 (cosine h - 1)/h = 0 and lim h->0 (sin h)/h = 1) are commonly used in calculus to evaluate certain types of limits.

Q: Why is it important to match the function f(a) with the expression in the derivative definition?

Matching the function f(a) with the expression in the derivative definition ensures that the correct derivative is obtained. In this case, it allows us to conclude that the derivative of sine is cosine.

Summary & Key Takeaways

• There are two ways to evaluate the limit of the expression as h approaches zero.

• In solution one, the derivative of a function at a is used to find the derivative of sine, which is cosine.

• In solution two, the limit is evaluated directly by applying trigonometric identities and simplifying the expression.