What is the equation of the tangent line of #f(x)=7x^2+3x# at #x=5#?

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Answer 1 · 2016-12-15T01:25:41

#y=73x-175 #

The gradient of the tangent to a curve at any particular point is give by the derivative of the curve at that point.

so If #y=7x^2+3x# then differentiating wrt #x# gives us:

#dy/dx = 14x+3#

When #x=5 => y=7*25+15=190# (so #(5,190)# lies on the curve)
and #dy/dx=14*5+3=73#

So the tangent we seek passes through #(5,190)# and has gradient #73# so using #y-y_1=m(x-x_1)# the equation we seek is;

# \ \ \ \ \ y-190=73(x-5) #
# :. y-190=73x-365#
# :. \ \ \ \ \ \ \ \ \ \ y=73x-175 #

We can confirm this solution is correct graphically:
enter image source here