Find the equation of the tangent line to the curve y=5xcosx at the point (pi,-5pi).The equation of this tangent line can be written in the form y=mx+b. Find m and b

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sciencesolve | Teacher | (Level 3) Educator Emeritus

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You need to remember how the equation of tangent line to a curve at a point looks like such that:

`y - f(x_0) = f'(x_0)(x - x_0)`

You need to identify what are the `x_0 ` and `f(x_0)`  coordinates of the point of tangency, hence, the problem provides that `x_0 = pi ` and `f(x_0) = -5pi.`

You need to differentiate the function with respect to x such that:

`f'(x) = (5x*cos x)'`

Notice that the function is a product of two factors, 5x and cos x, hence you need to use product rule such that:

`f'(x) = 5cos x- 5x*sin x`

You need to substitute `pi ` for x in `f'(x)`  to evaluate `f'(x_0) ` such that:

`f'(pi) = 5cos pi - 5pi*sin pi`

Since `cos pi = -1`  and `sin pi = 0`  yields:

`f'(pi) = -5`

You need to substitute `pi ` for `x_0` , `-5pi ` for `f(x_0) ` and `-5`  for `f'(x_0)`  in equation of tangent line such that:

`y+ 5pi= -5(x - pi)`

You need to use the slope intercept form of equation of tangent line, hence, you need to isolate y to the left side such that:

`y= -5x + 5pi - 5pi =gt y = -5x`

Hence, evaluating the equation of tangent line to the graph`y = 5xcos x` , at `(pi,-5pi)`  yields `y = -5x` .