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How do I find an equation of the tangent to the curve at the given point (using the chain rule)?
Ok I do not understand what I have to do. I tried looking at the book and I still do not understand. Maybe you can help me by giving me the steps to follow to do this: Here are some examples: a) y = (1 +2 x) ^ 10 (0.1) b) y = (x √ x ^ 2 +5) to (2.6) Note: x ^ 2 +5 to Apart from being part of the radical which means that its interior
To find the tangent at the given point, you need to find the slope at each point, So you must take a derivative. The purpose of the rule of the chain is making a differentiation using several simple steps: For example: dy / dx = (Dy / du) * (du / dx) a) Let u = (1 +2 x), then y = u ^ 10 ---> then you have: dy / du = 10u ^ 9, / dx = 2dx . Connect to our equation rule in the chain above, we have: dy / dx = 10u ^ 9 (2dx), re substitution in terms of x, we have: dy / dx = 20 (1 +2 x) dx * ^ 9 Evaluate our slope (0.1), we get: dy / dx = 20 (1 +2 (0)) ^ 9 = 20 Find the equation of the tangent to the curve in this area, use the form point slope (y-y_1) = m (x-x_1), using our point (0.1) and a slope of 20, this equation becomes: y-1 = 20 (x-0), simplifying yields: y = 20x ------- ------- 1 b) We need to use the product rule to find dy / dx: dy / dx = D (x) dx * sqrt (x ^ 2 +5) + x * d (sqrt [X ^ 2 +5]) dx We need to use the chain rule on the 2nd term to make a difference: Let f = sqrt (x ^ 2 +5) and u = (X ^ 2 +5), then: f = sqrt (u) or u ^ (2.1), then df / dx = (df / du) * (du / dx), then df / du = (1 / 2) ^ u (-1 / 2) or 1 / [2sqrt (u)] and du / dx = 2x * dx then: df / dx = 2x / [2sqrt (u)] = x / [sqrt (x ^ 2 +5)] Now, differentiating dy / dx, we get: dy / dx = sqrt (x ^ 2 +5) + x * x / [sqrt (x ^ 2 +5)] = sqrt (x ^ 2 +5) + x ^ 2 / [sqrt (x ^ 2 +5)] Now, plug in our point of find dy / dx (2.6) dy / dx (2.6) = Sqrt [(2) ^ 2 +5] + (2) ^ 2 / [sqrt ([2] ^ 2 +5)] = sqrt (9) + 4/sqrt (9) = 3 +1 = 4, using the point form slope of our information and data, we obtain: y-6 = 4 (x-2), the resolution of y yields: y = 4x ----- ------ 2 Hope this helps!
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