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).

We have been given the expression to be

Since we need to find the tangent at a point, we will have to find the derivative of 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

We will use the product rule of derivatives to find

Thus, (using the product rule which states that )

Taking the common factors out we get:

Thus, at is given by:

=Slope of the tangent of y at x=4=

Thus,

Now, the equation of the tangent line which passes through and has slope m is given by:

Thus, the equation of the tangent line which passes through and has the slope 185 is

Which can be simplified to

Thus,

This is the required equation of the tangent.

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Step-by-step explanation: