How do you find the equation of a line tangent to the function #y=3x-4sqrtx# at x=4?

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Answer 1 · 2016-09-23T03:34:18

#y=2x-4#

Find the point first where the tangent line will intersect on the curve:

#y(4)=3(4)-4sqrt4=4#

Implying the point #(4,4)#.

To find the slope of the tangent line, take the derivative of the function first, using the power rule: #d/dxx^n=nx^(n-1)#

#y=3x^1-4x^(1/2)#

#dy/dx=3(1)x^0-4(1/2)x^(-1/2)#

#dy/dx=3-2/sqrtx#

So, the slope at #x=4# is:

#dy/dx|_(x=4)=3-2/sqrt4=2#

Using the point #(4,4)# and slope #2# to write the tangent line:

#y-y_0=m(x-x_0)#

#y-4=2(x-4)#

#y=2x-4#

graph{(y-3x+4sqrtx)(y-2x+4)=0 [-1, 10, -5, 16]}