Why Use Logarithmic Differentiation in Calculus?
Mathematics is the abstract study of different topics such as quantity, number theory, structure (algebra). It is evolved from counting, measuring, and describing the shape of objects. The word Mathematics originates from the Greek word Mathema.
There are many branches of Mathematics such as algebra, geometry, trigonometry, calculus, probability, and statistics.
Logarithmic differentiation is a part of calculus. The technique is used in cases where it is easier to differentiate the logarithm of a function rather than the function itself. In some cases, it is easier to differentiate the logarithm of a given function than to differentiate it from the function itself.
It is a famous concept, and it applies to the majority of the non-zero functions. The only condition is that the non-zero functions should be differentiable in the future. Differentiation and integration are the two main concepts of calculus. Differentiation is used to study the small change of a quantity. Integration is different from differentiation. It is used to add small and discrete data and cannot be added singularly.
Derivative of Logarithm
When a logarithmic function is represented as:
f (x) = logb(x)
The derivative of a logarithmic function is given by:
f ' (x) = 1 / ( x ln(b) )
Here, x is called as the function argument.
b is the logarithm base.
ln b is the natural logarithm of b.
We can differentiate log in this way.
The derivative of ln(x) is 1/x
This is the way of differentiating ln. The derivative of ln(x) is a well-known derivative.
Following are some of the examples of logarithmic derivatives:
1. Find the Value of dy/dx if,y=ex4
Solution: Given the function y=ex4
Taking the natural logarithm of both the sides we get,
ln y = ln ex4
ln y = x 4 ln e
ln y = x4
Now, differentiating both the sides w.r.t we get,
1ydydx = 4x3
⇒dy dx =y.4x3
⇒dydx =ex4×4x3
2. Find the Value of dy dx if y = 2x{cos x}.
Solution: Given the function y = 2x{cos x}
Taking logarithm of both the sides, we get
log y = log(2x{cos x})
⇒log y=log2+log x cosx(As log(mn)=logm+logn)
⇒logy=log2+cosx×logx(As logmn=nlogm)
Now, differentiating both the sides w.r.t by using the chain rule we get,
1ydydx=cosx–(sinx)(logx)
This is a way used for differentiating logarithmic functions.
y ``(x)=(logax)′=1 xlna.
If a=e, we obtain the natural logarithm the derivative of which is expressed by the formula (lnx)′=1x,
where the number M is equal to M=log10e≈0.43429.
We derived the formula (logax)′=1xlna from first principles using the derivative's limit definition.
This is the way of deriving logarithms.
Deriving log functions becomes possible because of the use of exponents.
Following are some of the log derivative rules:
Common Functions
Function Derivative
Constant c 0
Line x 1
ax a
Square x2 2x
Square Root √x (½)x-½
Exponential ex ex
ax ln(a) ax
Logarithms ln(x) 1/x
loga(x) 1 / (x ln(a))
Trigonometry (x is in radians) sin(x) cos(x)
cos(x) −sin(x)
tan(x) sec2(x)
Inverse Trigonometry sin-1(x) 1/√(1−x2)
cos-1(x) −1/√(1−x2)
tan-1(x) 1/(1+x2)
Rules Function
Derivative
Multiplication by constant cf cf’
Power Rule xn nxn−1
Sum Rule f + g f’ + g’
Difference Rule f - g f’ − g’
Product Rule fg f g’ + f’ g
Quotient Rule f/g (f’ g − g’ f )/g2
Reciprocal Rule 1/f −f’/f2
Chain Rule
(as "Composition of Functions") f º g (f’ º g) × g’
Chain Rule (using ’ ) f(g(x)) f’(g(x))g’(x)
Chain Rule (using ddx ) dy dx = dy du du dx
These are logarithmic differentiation rules.
The logarithmic function with base a (a>0, a≠1) and exponential function with the same base form a pair of mutually inverse functions; the log function's derivative is also found using the inverse function theorem. (logax)′=f′(x)=1φ′(y)=1(ay)′=1aylna=1alogaxlna=1xlna.
The differentiation of natural log ln(x) is 1 divided by x.
Logarithmic differentiation steps are as follows:-
A natural log is supposed to be taken on both sides.
Use the property of the log of the product.
Differentiate on both sides. For every term on the right side of the equation, a chain rule should be used.
The last step is to multiply both sides by f(x).
Following are the logarithm derivative rules we always need to follow:-
The slope of a constant value (for example 3) is always 0.
The slope of a line like 2x is 2, or 3x is 3, etc.
One can use logarithmic differentiation when applied to functions raised to the power of variables or functions.
Logarithmic differentiation relies on the chain rule as well as the properties of the logarithm.
Differentiation in y
The derivative of function y = f(x) of a variable x is the measure of the rate at which the value y of the function changes concerning the change of the variable x. The derivative of ln y is 1/ (derivative of f = e\[^x\]) = 1/e\[^x\].
FAQs on Logarithmic Differentiation Explained: Formula, Steps & Examples
1. What is logarithmic differentiation, and when is it typically used?
Logarithmic differentiation is a special technique used to find the derivative of complex functions. It's most helpful for functions that have a variable in the exponent (like x^x) or involve the product or quotient of many functions. The method simplifies these problems by using the properties of logarithms before differentiating.
2. Can you explain the main steps involved in logarithmic differentiation?
The process is quite straightforward. Here are the typical steps:
- Start with the function, for example, y = f(x).
- Take the natural logarithm (ln) of both sides: ln(y) = ln(f(x)).
- Use the properties of logarithms to simplify the right-hand side. For example, break down products into sums or exponents into multipliers.
- Differentiate both sides with respect to x, using implicit differentiation on the left side (ln(y) becomes (1/y) * dy/dx).
- Solve for dy/dx to get the final derivative.
3. Which properties of logarithms are most important for this method?
The power of this method comes from three key logarithm properties:
- Product Rule: ln(a × b) = ln(a) + ln(b). This turns a complex product into a simple sum.
- Quotient Rule: ln(a / b) = ln(a) - ln(b). This turns a division problem into a simple subtraction.
- Power Rule: ln(a^b) = b × ln(a). This is crucial as it brings a variable exponent down, making it much easier to differentiate.
4. Why do we almost always use the natural logarithm (ln) for this process?
We use the natural logarithm (ln), which has a base of 'e', because its derivative is very simple. The derivative of ln(x) is just 1/x. If we used a different base, like log₁₀(x), its derivative would be (1/x) * (1/ln(10)), which adds an unnecessary constant and complicates the calculation. Using 'ln' keeps the process as clean and simple as possible.
5. How is logarithmic differentiation different from just using the chain rule or product rule?
While the chain rule and product rule are fundamental, they can become very complicated for certain functions. For example, differentiating a function like y = (sin x)^x is not possible with the standard power rule. Logarithmic differentiation simplifies these complex products, quotients, and variable exponents into manageable forms *before* you even start differentiating, making the overall process much easier.
6. What is a common mistake students make when using logarithmic differentiation?
A very common mistake is forgetting to use implicit differentiation correctly on the left side of the equation. When you differentiate ln(y) with respect to x, the answer is not simply 1/y. Since y is a function of x, you must use the chain rule, which gives (1/y) * (dy/dx). Many students forget the dy/dx part and then can't solve for the final derivative correctly.
7. Is logarithmic differentiation an important topic for the Class 12 CBSE Maths exam?
Yes, logarithmic differentiation is a key concept within the 'Continuity and Differentiability' chapter in the CBSE Class 12 Maths syllabus for 2025-26. It is frequently tested because it combines knowledge of logarithms, implicit differentiation, and standard differentiation rules. Questions on this topic often appear in the board exam.
8. Can this method be used for simple functions like y = x²?
Yes, it can be used, but it's not efficient. For y = x², you could take logs: ln(y) = 2ln(x). Differentiating gives (1/y)dy/dx = 2/x, which solves to dy/dx = 2x. While the answer is correct, it's much faster to use the simple power rule directly. This shows that logarithmic differentiation is a specialised tool for complex functions, not an everyday method.
