Calculus help on intro to the derivative.
For the function f'(x)=3^x, estimate f'(1). From the graph of f'(x), would you expect your estimate to be greater than or less than the true value of f'(1)?
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For f'(x) = `3^x`, the value of f'(1) can be estimated by substituting x by 1. That gives f'(1) = `3^1` = 3
The value of f'(1) obtained from the graph of f'(x), would be exactly equal to the estimated value of f'(1). This is verified from the graph of f'(x) = `3^x` that has been given below :
The graph passes through the point (1, 3) or f'(1) = 3.
The derivative of a function at a point is:
`f'(a) = lim_(h-gt0) (f(a+h)- f(a))/h`
`` For `a = 1 =gt f'(1) = lim_(h-gt0) (f(1+h)- f(1))/h`
`` You need to find f(x) integrating f'(x).
`int 3^x dx = (3^x)/ln3 + c`
`` `f'(1) = lim_(h-gt0) (3^(1+h) - 3^1)/(h*ln3)`
f'(1) = `3^1` = 3
The derivative of the function f(x) at the point x=1 is f'(1)=3.
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