at the point (0,0)
y = x / [√(25 + x²)]
y' = [√(25 + x²) * 1 - x * [x / √(25 + x²)] / (25 + x²)
y' = [√(25 + x²) - [x² / √(25 + x²)]] / (25 + x²)
y' = [[25 + x² - x²] / √(25 + x²)] / (25 + x²)
y' = 25 / [√(25 + x²) * (25 + x²)]
y' = 25 / [√(25 + x²)]³ → so that gives the slope at any point on the curve
When x = 0, y' = 25 / [√25 ]³ = 25/125 = 1/5 → slope at required point is 1/5
so the tangent passes thru (0, 0) with slope 1/5 and has the form y = mx + b
subsing stuff:
0 = (1/5) * 0 + b
b = 0
so the equation of the required tangent is y = (1/5)x
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y = x / [√(25 + x²)]
y' = [√(25 + x²) * 1 - x * [x / √(25 + x²)] / (25 + x²)
y' = [√(25 + x²) - [x² / √(25 + x²)]] / (25 + x²)
y' = [[25 + x² - x²] / √(25 + x²)] / (25 + x²)
y' = 25 / [√(25 + x²) * (25 + x²)]
y' = 25 / [√(25 + x²)]³ → so that gives the slope at any point on the curve
When x = 0, y' = 25 / [√25 ]³ = 25/125 = 1/5 → slope at required point is 1/5
so the tangent passes thru (0, 0) with slope 1/5 and has the form y = mx + b
subsing stuff:
0 = (1/5) * 0 + b
b = 0
so the equation of the required tangent is y = (1/5)x