How do you differentiate the curve y = 4x^2 + 7x + 1? And how do you find the gradient of this curve?
To begin with, this question requires you to differentiate the curve y = 4x2 + 7x +1 in order to find the gradient. To differentiate this function y in respect to x, we need to reduce the powers by one, for example in this question dy/dx (the gradient line) will become:dy/dx = (42)x2-1 + (71)x(1-1) + (10), which becomes dy/dx = 8x +7. This is the gradient of the curve, so in order to find the gradient of the curve at a specific point, we need to substitute the value we are given into dy/dx. For example, if you were asked to find the gradient of the curve at the point (1, 12), in this case x = 1 and y = 12, so when you subsitute x = 1 into dy/dx, the gradient = 15, as dy/dx = 81 +7. If the question was asking you to find the gradient when x = 5, dy/dx = 8*5 +7 = 47. Because this is a curve, the gradient is not the same at each point, as opposed to a straight line. Once you have found the value of dy/dx, you can use it to find the tangent to the curve at a point, or the normal (perpendicular to the tangent) to a curve at a given point.
