How do you use differentiation to find the slope of a curve?

You use differentiation to find the slope of a curve by calculating the derivative of its equation.

In more detail, the slope of a curve at any given point is essentially the gradient of the tangent line to the curve at that point. To find this gradient, you need to differentiate the function that represents the curve. Differentiation is a mathematical process that finds the rate at which a function is changing at any point.

For example, if you have a function \( y = f(x) \), the derivative of this function, denoted as \( \frac{dy}{dx} \) or \( f'(x) \), gives you the slope of the curve at any point \( x \). To differentiate a function, you apply specific rules such as the power rule, product rule, quotient rule, or chain rule, depending on the form of the function.

Let's say you have a simple function like \( y = x^2 \). To find the slope of the curve at any point, you differentiate \( y \) with respect to \( x \). Using the power rule, the derivative of \( x^2 \) is \( 2x \). This means the slope of the curve \( y = x^2 \) at any point \( x \) is \( 2x \). So, if you want to find the slope at \( x = 3 \), you substitute \( 3 \) into the derivative to get \( 2 \times 3 = 6 \).

Differentiation helps you analyse how a function behaves and changes, which is crucial in many areas of mathematics and science.