Chain Rule
The chain rule is a formula to compute the derivative of a composite function. If a variable x depends on the variable y, which in turn depends on the variable x, so that y = g(x) and z = f(y) = f(g(x)), then z depends on x via the intermediate variable y. The chain rule is expressed as:
dz/dx = (dz/dy) * (dy/dx)
Example
Let's say we have x = sin(y) and y = x^2. We want to find dz/dx.
1. Find dz/dy: The derivative of sin(y) with respect to y is cos(y).
2. Find dy/dx: The derivative of x^2 with respect to x is 2x.
3. Apply the chain rule: dz/dx = cos(y) * 2x. Since y = x^2, we substitute y back in: dz/dx = cos(x^2) * 2x.
Rate this AI answer:
Solving Linear Equations
Linear equations are fundamental in algebra. They involve variables raised to the first power and can be solved using basic algebraic operations. This guide will walk you through the steps to solve linear equations effectively.
Step 1: Simplify Both Sides
If either side of the equation has terms that can be combined or simplified, do so first. This might involve distributing terms or combining like terms.
Step 2: Isolate the Variable Term
Use addition or subtraction to move all terms involving the variable to one side of the equation and all contant terms to the other side.
Step 3: Solve for the Variable
If the variable term has a coefficient other than 1, divide both sides of the equation by that coefficient to solve for the variable.