When x here is replaced with something else here, say, d/dx(sin 3x) =? In such cases, we apply the chain rule formula which says d/dx ( f(g(x) ) = f' (g(x))
Usually, all the derivative formulas are in terms of x, for example, d/dx (sin x) = cos x. There are two forms of the chain rule formula. The chain rule formula is used to find the derivative of a composite function (i.e, when one function is inside the other). If there exists a function f of g which in turn is a function of u(x), then the instantaneous change in f with respect to x is given as change in f/ change in x = change in g /change in u × change in u /change in x. The chain rule is used to find the derivative of a composite function. Thus the equation of the tangent line to the given function y = (5 x 4 - 2) 3 applying the chain rule formula is y = 540x - 513 Hence substitute (1,27) in the equation of the tangent line, y = 540x + b, we get We need to find the equation of the tangent line. Therefore the equation of the tangent line in the slope-intercept form is y = mx+ b ⇒ 27 = 540x + b We need to evaluate the function and the derivative at the given point We know that the derivative of the function gives the slope of the line at the given point. Let us apply the chain rule to find the equation of the tangent line to the given function y = (5 x 4 - 2) 3 at x = 1. To find the rate of change of the average molecular speed,.To determine if a function is increasing or decreasing,.To find the position of an object that is moving to the right and left in a particular interval,.To calculate the rate of change of distance between two moving objects,.To find the time rate of change of the pressure,.This chain rule has broad applications in the fields of physics, chemistry, and engineering. 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.
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