Part a
Differentiate \(f(x)\) for each piece: For \(x < 1\), \(f(x) = 2x + 3\); for \(x \geq 1\), \(f(x) = x^2\). Provide the derivative expressions for each interval.
Implicit Differentiation of a Quadratic Curve
Consider the curve defined by \(x^2 + xy + y^2 = 4\). (a) For SL students: Differentiate implicitly to find \(\frac{dy}{dx}\) in terms of \(x\) and \(y\). (b) For HL extension: Using your result from part (a), determine the slope of the tangent line at the point where \(x = 1\) and find the corresponding […]
Logarithmic Differentiation of Exponential Functions
Differentiate the function \(f(x)= x^x\) with respect to \(x\) using logarithmic differentiation.
Differentiation of the Sine Function
Differentiate the function \(f(x) = \sin(x)\) with respect to \(x\).