Find the equation of the tangent to the curve y = x√x at the point (4,6).

Simply the equation. y = x√x is the same as y = x^3/2. This is because the square root of any articular number is the same as saying that number raised to the power half. Now to do x¹ multiply x^1/2 we simply add the powers together to get 3/2. Differentiate the curve. dy/dx = (x^3/2) multiply by 3/2 and subtract 1 from the power = 3/2x^1/2. In differentiation we multiply by the whole expression by the power and then subtract the power by 1. Substitute the x-coordinate into the dy/dx equation. When x = 4, dy/dx = 3/2 multiply by 4^1/2. To do this we first do 4^1/2, we discussed earlier that any number raised to the power half is the same as finding the square root of that number. In this case, the square root of 4 is 2. Next you multiply 2 by 3/2 which is 3. The number 3 is the gradient of the tangent to the curve. So the equation of the tangent so far is y = 3x + c. Since we know the y-coordinate we can find the y-intercept (c) by substituting the coordinates (4,6) into the equation. So when x = 2, y has to equal 6.6 = (3 x 4) + c. So 3 x 4 is 12, 6 = 12 + c Therefore c = -6. The tangent to the curve is therefore y = 3x - 6.