Simple Example of Derivatives

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Implicit Differentiation (An example)

Q Differentiate

A A trick to do this question is to convert the question to . Now do implicit differentiation to get . Since as know , then . (there're two ways to get this. One way is to use trig identity ; the other way is to draw a right-angled triangle with angle , adjacent side and hypotenuse . Then the opposite side is , and . So going back to the derivative and isolate .

Q Difference between secant and a tangent:

A A secant is any line passing through a graph. It would intersect the graph at least two points. However when two such points get closer and closer to each other, the secant becomes a tangent to a graph. One particular application is finding a derivative or slope of tangent at certain point. First find slope of the secant at and and then take limit

Q Find the derivative of using the definition of limits

A

Q find the tangent to the graph f(x) at x=a

A slope of tangent = f'(x), equation of tangent line is [using the formula ]

Q Find the derivative of

A Using the product rule we can write .

Now the derivative of will be using chain rule

Derivative of using chain rule

So

Hint to find derivative of : Derivative of is using chain rule. use this formula . An extra will come by derivative of

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