What is the equation of the tangent line of #f(x) =2x^3+6x^2+2x3# at #x=3#?
y =  16x + 51
The line's equation is y  b = m (x  a), where (a, b) is a line point and m is the gradient.
We differentiate f(x) to obtain the gradient m (gradient), since f'(x) provides the gradient of the tangent to the curve.
= 3  54 + 54 + 6 =
The tangent equation is y  b = m(x  a), where m =  16 and (a, b ) = (3, 3 ).
Consequently, y  3 =  16(x  3) and y  3 =  16x + 48.
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The equation of the tangent line of f(x) = 2x^3 + 6x^2 + 2x  3 at x = 3 is y = 45x  99.
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To find the equation of the tangent line to the function ( f(x) = 2x^3 + 6x^2 + 2x  3 ) at ( x = 3 ), follow these steps:
 Find the slope of the tangent line by taking the derivative of the function.
 Find the value of the function at ( x = 3 ) to get the ( y )coordinate of the point of tangency.
 Use the pointslope form of the equation of a line to write the equation of the tangent line.
Let's go through these steps:

Find the derivative of the function: [ f'(x) = 6x^2 + 12x + 2 ]

Evaluate the derivative at ( x = 3 ): [ f'(3) = 6(3)^2 + 12(3) + 2 ] [ f'(3) = 54 + 36 + 2 ] [ f'(3) = 16 ]

The slope of the tangent line is ( m = 16 ). Now, find the ( y )coordinate of the point of tangency by plugging ( x = 3 ) into the original function: [ f(3) = 2(3)^3 + 6(3)^2 + 2(3)  3 ] [ f(3) = 54 + 54 + 6  3 ] [ f(3) = 3 ]
So, the point of tangency is ( (3, 3) ).
 Now, use the pointslope form of the equation of a line to find the equation of the tangent line: [ y  y_1 = m(x  x_1) ] [ y  3 = 16(x  3) ] [ y  3 = 16x + 48 ] [ y = 16x + 51 ]
Therefore, the equation of the tangent line to ( f(x) = 2x^3 + 6x^2 + 2x  3 ) at ( x = 3 ) is ( y = 16x + 51 ).
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When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
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