# Eg find the derivative with respect to v for 2 1 3 5

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e.g. Find the derivative with respect to v for 2 1 3 5 z v e.g. Find 1 ( ) if ( ) 2 d h t h t dt t e.g. Find the slope of the tangent line of the function at the given point: a. ( ) 2 at (7,3) f x x b. 2 ( ) 4 at (2, 4) f x x x
5 e.g. Find the equation of the tangent line to the graph at the given point: a. 2 ( ) 3 at (1,2) f x x x c. 1 1 ( ) at ,2 2 f x x
Alternate form of the derivative : ( ) ( ) ( ) lim x c f x f c f c x c provided the limit exists. e.g. Is the function 2 3 ( ) f x x differentiable at 0? e.g. Use the alternative form on the derivative to find the derivative at x c , if it exists. x
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7 f is differentiable on [ a , b ] if it is differentiable on ( , ) a b , and ( ) ( ) x a f x f a x a and ( ) ( ) lim x b f x f b x b exist. We call these one-sided limits the derivatives from the right and left. e.g. Is the function 2 ( ) f x x differentiable on [0, 5] ? Note: Functions are not differentiable where there is a sharp turn or where they have a vertical tangent line. e.g. Identify any points at which the function is not differentiable: a. b. Theorem If f is differentiable at x c , then f is continuous at x Note: If f lim c .