How do you find an equation at a tangent line to the curve y=(5+4x)2 at point p = (7, 4)?

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Morgan Griffin · Accepted Answer

To find the equation of the tangent line to the curve

y = {(5 + 4 x)}^{2}

at point (7,4) we need to find the gradient/slope of the tangent line at the point (7,4) to do this, we need to differentiate our function.
Note that the point does not exist on that curve, so I will use a point that does exist on the curve, which would be (7, 1089)
I got to that point by simply setting x=7 in the equation to get y

\frac{d y}{d x} f (x) = \frac{d y}{d x} {(5 + 4 x)}^{2}

f'(x)=8(5+4x)
then we sub in the x value from the given point.
f'(7)=8(5+4(7))=264
now we have got the gradient or m at the point where x=7
we can start using our formula for a straight line, which is:
y=mx+c
and we need to solve for c so we use our known point.
1089=(264)(7)+c
c=-759
thus the equation to the line tangent to the curve at the point where x=7 would be.
y=264x-759

How do you find an equation at a tangent line

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