The chain rule is used to differentiate composite functions (functions within functions). It states that d/dx [f(g(x))] = f(g(x)) * g(x). Your child should use it when differentiating functions like sin(x^2) or e^(3x).
The product rule is used to differentiate the product of two functions. It states that d/dx [u(x)v(x)] = u(x)v(x) + u(x)v(x). Your child should use it when differentiating functions like x*sin(x) or e^x*ln(x).
The quotient rule is used to differentiate a function that is the ratio of two functions. It states that d/dx [u(x)/v(x)] = [v(x)u(x) - u(x)v(x)] / [v(x)]^2. Use it when differentiating functions like sin(x)/x or (x^2 + 1)/e^x.
The derivatives of basic trigonometric functions are: d/dx [sin(x)] = cos(x), d/dx [cos(x)] = -sin(x), and d/dx [tan(x)] = sec^2(x). Remember to apply the chain rule if the argument of the trigonometric function is not simply x.
Implicit differentiation is used when you have an equation where y is not explicitly defined as a function of x (e.g., x^2 + y^2 = 1). Differentiate both sides of the equation with respect to x, treating y as a function of x and using the chain rule when differentiating terms involving y. Then, solve for dy/dx.
The derivative of e^x is e^x. The derivative of ln(x) is 1/x. For other exponential functions like a^x, the derivative is a^x * ln(a). Remember to use the chain rule if the exponent or argument is a function of x.
Parametric equations define x and y in terms of a third variable, usually t (e.g., x = t^2, y = sin(t)). To find dy/dx, calculate dy/dt and dx/dt, then dy/dx = (dy/dt) / (dx/dt).
To find the second derivative, differentiate the first derivative (dy/dx) with respect to x. Remember to use the chain rule if necessary. The second derivative tells you about the concavity of the function: positive means concave up, negative means concave down.
Stationary points are points where the derivative (dy/dx) is equal to zero. These points can be local maxima, local minima, or points of inflection. To find them, set dy/dx = 0 and solve for x. Then, use the second derivative test to determine the nature of the stationary point.
Optimization problems involve finding the maximum or minimum value of a function subject to certain constraints. Use differentiation to find the critical points (where the derivative is zero or undefined). Then, use the first or second derivative test to determine whether each critical point is a maximum or minimum. Consider the constraints to find the absolute maximum or minimum within the given interval.