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Mamont248 [21]
3 years ago
10

Find the equation of the tangent line to the curve y equals left parenthesis x minus 3 right parenthesis superscript 7 baseline

left parenthesis x plus 1 right parenthesis squaredy=(x−3)7(x+1)2 at the point left parenthesis 4 comma 25 right parenthesis(4,25).
Mathematics
1 answer:
Rashid [163]3 years ago
3 0

We have been given the expression to be y=(x-3)^7(x+1)^2

Since we need to find the tangent at a point, we will have to find the derivative of y as the slope of the tangent at a given point on the curve is always equal to value of the derivative at that point.

Thus, we have to find \frac{dy}{dx}=\frac{d}{dx} (x-3)^7(x+1)^2

We will use the product rule of derivatives to find \frac{dy}{dx}

Thus, y'=7(x-3)^6(x+1)^2+(x-3)^7\times2(x+1) (using the product rule which states that (fg)'=f'g+fg')

Taking the common factors out we get:

y'=(x-3)^6(x+1)(7(x+1)+2(x-3))

y'=(x-3)^6(x+1)(7x+7+2x-6)=(x-3)^6(x+1)(9x+1)

Thus, y' at x=4 is given by:

y'_{x=4}=Slope of the tangent of y at x=4=m

Thus, m=(4-3)^6(4+1)(9\times4+1)=185

Now, the equation of the tangent line which passes through (x_{1}, y_{1}) and has slope m is given by:

y-y_{1}=m(x-x_{1})

Thus, the equation of the tangent line which passes through (4,25) and has the slope 185 isy-25=185(x-4)

Which can be simplified to y=185x-740+25=185x-715

Thus, y=185x-715

This is the required equation of the tangent.

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