If f(x)=2x^(5/2), find f'(4). In the equations of a tangent line m is and b is?

Use orginal equations f(x)=2x^(5/2), find f'(4). using that equation find the equation of the tangent line to the curve y=2x^(5/2) at the point (4,f(4)). The equation of this tangent line can be written in the form y=mx+b

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Given `y=2x^(5/2)` :

(1) `f'(x)=(2)(5/2)(x^(3/2))=5x^(3/2)`

(2) `f'(4)=5(4)^(3/2)=5*8=40` This is the slope of the tangent line to f(x) at x=4.

(3) `f(4)=2(4)^(5/2)=2*32=64` . Thus the point (4,64) lies on the graph of f(x).

(4) We have the slope m=40 and a point (4,64). The equation of the line is `y-64=40(x-4)` or `y=40x-96`

**The equation of the tangent line to `y=2x^(5/2)` at x=4 is y=40x-96**

The graph of f(x) and the tangent line at x=4:

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