We can assume the expression that is replacing 'x' with 'u' and applying the chain rule formula.Įxample : To find d/dx (sin 2x), assume that y = sin 2x and 2x = u. Chain Rule Formula 1:Įxample : To find the derivative of d/dx (sin 2x), express sin 2x = f(g(x)), where f(x) = sin x and g(x) = 2x. There are two forms of chain rule formula as shown below. On simplifying we get, cos x/sin x = cot x.Finally g'(x) = derivative of the outside function, leaving the inside alone × the derivative of the inside function = 1/sin x × cos x.The derivative of the inner function is cos x.The derivative of the outer function is 1/sin x.sin x is the inner function and ln(x) is the outer function.Step 6: Simplify the chain rule derivative.įor example: Consider a function: g(x) = ln(sin x).Step 5: Multiply the results from step 4 and step 5.Step 4: Find the derivative of the inner function.Step 3: Find the derivative of the outer function, leaving the inner function.Step 2: Identify the inner function and the outer function.Step 1: Identify The Chain Rule: The function must be a composite function, which means one function is nested over the other.This is given as Leibniz notation in the form of an equation as df/dx = dg/du. Suppose u = h(x), where du/dx and dg/du exist, then this could be expressed as:Ĭhange in f/ change in x = change in g /change in u × change in u /change in x. The Theorem of Chain Rule: Let f be a real-valued function that is a composite of two functions g and h.

It is used only to find the derivatives of the composite functions. This chain rule is also known as the outside-inside rule or the composite function rule or function of a function rule.

Chain rule calculus with steps