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#CHAIN RULE CALCULUS PDF HOW TO#

We bring down the 4 in front of the brackets, the power becomes 3 and we keep sin(2𝑥) inside the brackets, which gives us 4 3. Bring down the power and subtract one from the power, keeping the trigonometric function the same. Multiply this by the derivative of the trigonometric function.įor example, differentiate sin 4(2𝑥) using the chain rule.Bring down the power and subtract one from the power, keeping the trigonometric function inside the same.Write the trigonometric function as the inner function in brackets and the power as the outer function.The chain rule can be applied to trigonometric functions raised to a power. The inner function is 2𝑥 and its derivative is 2. The following rules for differentiating the trigonometric functions of sin(𝑥), cos(𝑥) and tan(𝑥) may be useful here: For example, differentiate cos(2𝑥) using the chain ruleĬos(2𝑥) can be written as an inner function of 2𝑥 and an outer function of cos().Ĭos(𝑥) differentiates to -sin(𝑥) and so, keeping the inner function as 2𝑥 rather than 𝑥, we get -sin(2𝑥). Differentiate the trigonometric function, keeping the inner function the same and then multiply this by the derivative of the inner function. The chain rule is used to differentiate trigonometric functions containing another function. The chain rule formula states that F'(𝑥) = f'(g(𝑥).g'(𝑥), where g(𝑥) is the inner function and f(𝑥) is the outer function. We will now use the chain rule formula to differentiate this function.

So we must multiply the result of 5(4𝑥 – 3) 4 by 4. The inner function is the function inside the brackets. we already have 5(4𝑥 – 3) 4 and now we must multiply this by the derivative of the inner function. Multiply this by the derivative of the inner functionįrom step 1. However because we have 4𝑥 – 3 inside the brackets and not just 𝑥, we must also include step 2. We keep the inner function of 4𝑥-3 the same, so we write 5(4𝑥 – 3) 4. 𝑥 5 would differentiate to 5𝑥 4 and so we write ( ) 5 differentiated as 5( ) 4. We differentiate this like we would 𝑥 5. Differentiate the outer function, keeping the inner function the same We define 4𝑥 – 3 as the inner function and the ( ) 5 as the outer function.

Below this, we will use the chain rule formula method. In this example we will use the chain rule step-by-step.

Multiply this by the derivative of the inner function.įor example, differentiate (4𝑥 – 3) 5 using the chain rule.

Differentiate the outer function, keeping the inner function the same.

#CHAIN RULE CALCULUS PDF HOW TO#

How to Do the Chain Rule To do the chain rule: Such functions must be differentiable themselves.

The function must be a composite function of two or more functions.

To use the chain rule, the following rules are required: In words, the chain rule requires finding the derivative of the outer function while keeping the inner function the same and then multiplying this by the derivative of the inner function. g(𝑥) is the inner function and f(𝑥) is the outer function. The chain rule is defined as, where u is a function of 𝑥 ( u = g(x) ) and y is a function of u ( y = f(u) ).Īlternatively, the chain rule can be written in function notation as F'(𝑥) = f'(g(𝑥)).g'(𝑥), where F(𝑥) = f(g(𝑥)). The chain rule is used when a function is within another function.