I have used first principles for x^n , ln x, cos x, sin x . the derivatives of all other elementary functions can then be calculated using the product rule and the chain rule alone . Even arbitrary powers x^p (p real ) can be calculated in this way if only the derivative of f(x) = x is known.
I have seen many videos in which the derivative of some function is derived from first principles . I don't think that this is the way to calculate derivatives . If you know linearity and the product rule you can get e.g. the derivative of any function containing (positive or negative ) powers of x. Assume you know that x' =1 (this is trivial indeed ) .Then , (x^2 ) ' = (x * x )' = 2*x ' *x = 2 x. Another example : f(x) =√x . We may now use that (f[x] *f[x]) ' = 1 = 2*f[x] ' f[x] -> f[x] ' =1 / (2 √x ) . The quotient rule can be derived with this same trick , and you can get a list of derivatives with this procedure.
Your wrong that is exactly the way to calculate derivatives, doing it your way doesn’t show conceptually what a derivative is. Remember the derivative is the slope of the tangent line at any point. Using the difference quotient shows people who are new to calculus that this is merely rise over run which is slope which people have been learning since grade school. The only difference is the modification of h approaching 0 which makes the secant line shorter and shorter until it becomes a tangent line which gives the slope at that exact point.
@@renesperb I didn’t say you didn’t know what a derivative is. I’m saying that this is the way to calculate using the difference quotient because that’s how a derivative is defined. Your method is easier to get to the solution faster, however first principals is how we got started.
