Differentiation

Principles of Differentiation

• Grasp the concept of differentiation, which describes how a function changes as its input changes.
• Understand the derivative of a function as the slope of the tangent to that function at any point.
• Comprehend the derivative of a constant function as zero, since the slope of a line parallel to the x-axis is zero.
• Perceive the geometric interpretation of the derivative as the instantaneous rate of change.
• Be familiar with the concept of higher order derivatives, which are the derivatives of derivatives, expressing acceleration, concavity, and other complex changes.

Rules of Differentiation

• Acknowledge the power rule for differentiation which states that if y = ax^n, then dy/dx = nax^(n-1).
• Realise the sum rule which states that the derivative of the sum of two functions is the sum of their derivatives.
• Understand the product rule which expresses the derivative of the product of two functions: if y = f(x)g(x), then dy/dx = f’(x)g(x) + f(x)g’(x).
• Familiarise with the chain rule, crucial for differentiating composite functions: dy/dx = dy/du * du/dx, where u is a function of x.
• Grasp the quotient rule which states that if y = f(x) / g(x), then dy/dx = [f’(x)g(x) - g’(x)f(x)] / (g(x))^2.

Differentiation Applications

• Know how to find equations of tangents and normals to curves by using derivatives.
• Understand how to apply differentiation to locate and classify turning points of curves.
• Realise how to use the second derivative test to classify stationary points as maxima, minima or points of inflection.
• Comprehend the concept of optimization problems where differentiation is used to find maximum or minimum values.
• Acknowledge the use of differentiation in kinematics - calculating velocity and acceleration given displacement as a function of time.
• Be able to use differentiation to solve rate of change problems in a variety of real-world contexts.