Chain Rule
- Introduction
- What Is Chain Rule?
- Chain Rule Steps
- Chain Rule Formula
- Double Chain Rule
- Applications of Chain Rule
- Solved Examples
- Practice Problems
- Frequently Asked Questions
Introduction
When we have a function that is a composition of two or more functions, we use the chain rule to differentiate it. In calculus, the chain rule is used to calculate derivatives of composite functions such as `e^(2x)`, `(x^2 + 1)^3`, `(sin 2x)`, `(ln 5x)`, and more. To get the derivative of a composite function `y = f(g(x))`, we basically use the chain rule formula, which is `y' = f'(g(x)) * g'(x)`. We take into account the rats of change of `f` with respect to `g` and `g` with respect to `x`, to calculate calculating the rate of change of `f` with respect to `x`. We can quickly compute the derivatives of complex functions by using the chain rule.
What Is Chain Rule?
Chain rule states that the derivative of a composite function is the derivative of the outer function with respect to the inner function times the derivative of the inner function. Chain rule, also called the outside-inside rule or the composite function rule, is significantly used in calculus, to determine the derivatives of composite functions. It aids in figuring out how modifications to one function's input impact another function's output. In essence, it enables us to distinguish one function from another. This formula is important in many branches of mathematics, but it's especially helpful in engineering, physics, and optimization. We can solve motion, growth, and optimization problems as well as assess the rate of change of complex functions by using the chain rule.
Chain Rule Steps
To apply the chain rule, we basically follow the following steps in order.
`1`. First, identify that the function is composed of nested functions.
`2`. Then, identify the inner function and the outer function.
`3`. Next, find the derivative of the outer function while leaving the inner function unchanged.
`4`. After that, find the derivative of the inner function. Multiply the results from these derivatives together, and finally, simplify the expression.
Chain Rule Formula
Alternatively, if `y` is a function of `u`, and `u` is a function of `x`, then as per the chain rule.
Example `1`: Find the derivative of cos (2x).
Solution:
To find `d/dx (cos (2x))`, assume that `y = cos (2x)` and `2x = u`. Then `y = cos (u)`.
Then by the chain rule formula,
Therefore: `(du)/(dx) = 2` and `(dy)/(du) = −sin (u)` and so, the chain rule says:
`(dy)/(dx) = (dy)/(du)*(du)/(dx)`
`(dy)/(dx) = −sin (u)*(2)`
`(dy)/(dx) = -2 sin (2x)`
Thus the derivative of `y` with respect to `x` will be `-2 sin (2x)`.
Example `2`: Differentiate `sin (2x)`.
Solution:
To find `d/dx (sin (2x))`, assume that `y = sin (2x)` and `2x = u`. Then `y = sin (u)`.
Then by the chain rule formula,
Therefore: `(du)/(dx) = 2` and `(dy)/(du) = cos (u)` and so, the chain rule says:
`(dy)/(dx) = (dy)/(du)*(du)/(dx)`
`(dy)/(dx) = cos(u)*(2)`
`(dy)/(dx) = 2 cos (2x)`
Thus the derivative of `y` with respect to `x` will be `2 cos (2x)`.
Example `3`: Differentiate `cos x^2`.
Solution:
To find `d/dx (cos x^2)`, we will apply the chain rule as follows.
`y = cos (x^2)` Let `u=x^2`, then we have `y=cos (u)`
Therefore: `(du)/(dx) = 2x` and `(dy)/(du) = −sin (u)` and so, the chain rule says:
`(dy)/(dx) = (dy)/(du)*(du)/(dx)`
`(dy)/(dx) = −sin(u)×2x`
`(dy)/(dx) = –2xsin(x^2)`
Thus the derivative of `y` with respect to `x` will be `−2xsin(x^2)`.
Double Chain Rule
The chain rule is a fundamental concept in calculus that deals with nested functions. Imagine you have several functions nested within each other, where each function depends on more than one variable. In such cases, the chain rule tells us how to find the derivative of the outermost function with respect to one of its variables. For example, let's say we have three functions: `u`, `v`, and `w`, and a function `f` that is a composition of these three. In this scenario, we apply the chain rule multiple times.
If `f=(u\circ v)\circ w`, then to find the derivative of `f` with respect to `x`, we would multiply the derivatives of each nested function together: `(df)/(dx) = (df)/(du) * (du)/(dv) * (dv)/(dw) * (dw)/(dx)`. This process allows us to calculate the overall derivative of the composite function.
When applying the chain rule, we don't necessarily need to recall the specific formula. Instead, we can utilise the derivative formulas in terms of `x` and then multiply the result by the derivative of the expression replacing `x`.This approach allows us to find derivatives efficiently without needing to memorise the chain rule formula.
Applications of Chain Rule
The chain rule is widely applicable in various fields like physics, chemistry, and engineering. It allows us to find derivatives by simply applying the derivative formulas and then multiplying by the derivative of the expression that replaces `x`. This rule is used to:
- Determine the rate of change of pressure over time,
- Calculate the rate of change of distance between two moving objects,
- Find the position of an object moving in different directions within a given interval,
- Determine whether a function is increasing or decreasing,
- Calculate the rate of change of average molecular speed.
Solved Examples
Example `1`: Find the derivative of the function `f(x)=(2x^3+5x)^4` with respect to `x`.
Solution:
To do this, we'll use the chain rule, which states that if `y = f(u)` and `u = g(x)`,
Then `(dy)/(dx)= (dy)/ (du) * (du) / (dx)`
First, we identify `u` as the inner function:
`u = 2x^3+5x`
Now, we'll differentiate `u` with respect to `x`
`(du)/(dx) = 6x^2+5`
Next, we differentiate `f(u)= u^4` with respect to `u`
`(df)/(du)= 4u^3`
Now applying chain rule, we get
`(df)/(dx)= (df)/(du) * (du)/(dx)`
`= (4u^3)*(6x^2+5)`
Substituting the value of `u`, we get
`=4((2x^3+5x)^3)*(6x^2+5)`
Example `2`: Find the derivative of `f(x)= sin(3x^2+2x)` with respect to `x`.
Solution:
Let `u = 3x^2+2x` and `f = sin(u)`
Now using chain rule,
`(du)/(dx)= d/dx(3x^2+2x)= 6x+2`, and `(df)/(du)= d/(du)(sin(u))= cos u`
Using chain rule,
`(df)/(dx)= (df)/(du). (du)/(dx) =cos(u)*(6x+2)`
Substituting `u = 3x^2+2x`, we get
`(df)/(dx)= cos(3x^2+2x)* (6x+2)`
Hence, `(df)/(dx)= cos(3x^2+2x).(6x+2)`
Example `3`: Let \( g(x) = \sqrt{4x^2 + 1} \). Find \( g'(x) \).
Solution:
Applying the chain rule:
`g'(x) = \frac{1}{2\sqrt{4x^2 + 1}} \cdot \frac{d}{dx}(4x^2 + 1)`
`= \frac{1}{2\sqrt{4x^2 + 1}} \cdot 8x`
`= \frac{4x}{\sqrt{4x^2 + 1}}`
Example `4`: If \( h(x) = e^{2x^3 - 5x} \), find \( h'(x) \).
Solution:
Applying the chain rule:
`h'(x) = e^{2x^3 - 5x} \cdot \frac{d}{dx}(2x^3 - 5x)`
`= e^{2x^3 - 5x} \cdot (6x^2 - 5)`
Example `5`: Let \( y = (2z + 3)^4 \). Find \(\frac{dy}{dz}\) using the chain rule.
Solution:
Using the chain rule:
`\frac{dy}{dz} = 4(2z + 3)^3 \cdot \frac{d}{dz}(2z+ 3)`
`= 4(2z + 3)^3 \cdot 2`
`= 8(2z + 3)^3`
Practice Problems
Q`1`. What is the derivative of \( f(x) = \sin(3x) \) with respect to \( x \)?
- \( 3\cos(3x) \)
- \( \cos(3x) \)
- \( \sin(3x) \)
- \( 3\sin(3x) \)
Answer: a
Q`2`. If \( g(x) = \sqrt{5x^2 + 3} \), what is \( g'(x) \)?
- `\frac{5x}{\sqrt{5x^2 + 3}}`
- `\frac{5x}{2\sqrt{5x^2 + 3}}`
- `\frac{5x}{2\sqrt{5x^2 - 3}}`
- `\frac{5x}{\sqrt{5x^2 - 3}}`
Answer: b
Q`3`. If \( h(x) = e^{2x^2} \), what is \( h'(x) \)?
- \( 4xe^{2x^2} \)
- \( 4x^2e^{2x^2} \)
- \( 2xe^{2x^2} \)
- \( e^{2x^2} \)
Answer: a
Q`4`. Given \( y = \ln(3x^2 + 4) \), what is `\frac{dy}{dx}`?
- `\frac{6x}{3x^2 - 4}`
- `\frac{6x}{2(3x^2 + 4)}`
- `\frac{6x}{3x^2 + 4}`
- `\frac{6x}{2(3x^2 - 4)}`
Answer: c
Q`5`. If \( f(x) = \cos(2x^2 + 1) \), what is \( f'(x) \)?
- \( -4x\sin(2x^2 + 1) \)
- \( -4x\sin(2x^2) \)
- \( -4x^2\sin(2x^2 + 1) \)
- \( -4x\sin(4x) \)
Answer: a
Frequently Asked Questions
Q`1`. What is the chain rule in calculus?
Answer. The chain rule is a fundamental theorem in calculus that describes how to find the derivative of a composite function. It states that if `f` and `g` are differentiable functions, then the derivative of their composition `f(g(x))` is given by `(fog)'(x)=f'(g(x))*g'(x)`
Q`2`. When should I use the chain rule?
Answer. You should use the chain rule when you need to find the derivative of a function that is composed of two or more functions nested within each other. For example, when you have functions like `sin(x^2)`, you will need to apply the chain rule
Q`3`. How do I apply the chain rule?
Answer. To apply the chain rule, you first identify the outer function and its derivative, then the inner function and its derivative. Next, you differentiate the outer function with respect to the inner function, and multiply by the derivative of the inner function with respect to the variable of differentiation.
Q`4`. Can the chain rule be used with more than two functions?
Answer. Yes, the chain rule can be extended to functions composed of more than two functions. If you have a composition of three or more functions, you apply the chain rule successively, starting from the outermost function and working inward.
Q`5`. Are there any common mistakes to avoid when using the chain rule?
Answer. One common mistake is forgetting to multiply by the derivative of the inner function when applying the chain rule. Another mistake is incorrectly identifying the inner and outer functions. It's important to properly recognize the composition of functions to apply the chain rule correctly. Additionally, be careful with the algebraic manipulation when simplifying derivatives obtained using the chain rule.