Logarithmic Differentiation Formula Steps and Solved Examples for Competitive Exams
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 and How to Differentiate Complex Functions
1. What is logarithmic differentiation?
Logarithmic differentiation is a technique used to differentiate functions by first taking the natural logarithm of both sides and then applying implicit differentiation. It is especially useful when dealing with:
- Products of many factors
- Quotients of complex expressions
- Functions where both base and exponent contain variables, such as x^x
2. When should you use logarithmic differentiation?
You should use logarithmic differentiation when the function is complicated to differentiate directly, especially if it involves products, quotients, or variable exponents. It is particularly helpful for:
- y = (x^2 + 1)^5 (x^3 − 2)^4
- y = x^x
- Multiple factors multiplied or divided together
3. How do you do logarithmic differentiation step by step?
To perform logarithmic differentiation, take the natural log of both sides and then differentiate implicitly. Follow these steps:
- Step 1: Let y = f(x)
- Step 2: Take ln on both sides: ln y = ln f(x)
- Step 3: Use log rules to simplify
- Step 4: Differentiate both sides with respect to x
- Step 5: Solve for dy/dx
4. How do you differentiate x^x using logarithmic differentiation?
The derivative of x^x using logarithmic differentiation is x^x(1 + ln x). Steps:
- Let y = x^x
- Take ln: ln y = x ln x
- Differentiate: (1/y) dy/dx = ln x + 1
- Multiply by y: dy/dx = y(ln x + 1)
5. What is the formula for logarithmic differentiation?
The general idea of logarithmic differentiation is based on the identity d/dx[ln y] = (1/y) dy/dx. If y = f(x), then:
- ln y = ln f(x)
- (1/y) dy/dx = d/dx[ln f(x)]
- dy/dx = y · d/dx[ln f(x)]
6. Why does logarithmic differentiation work?
Logarithmic differentiation works because logarithms convert products into sums and powers into multipliers, making derivatives easier to compute. Key logarithm rules used are:
- ln(ab) = ln a + ln b
- ln(a/b) = ln a − ln b
- ln(a^n) = n ln a
7. Can you give an example of logarithmic differentiation with a product?
Yes, logarithmic differentiation simplifies products like y = (x^2 + 1)(x^3 − 2). Steps:
- Take ln: ln y = ln(x^2 + 1) + ln(x^3 − 2)
- Differentiate: (1/y) dy/dx = (2x)/(x^2 + 1) + (3x^2)/(x^3 − 2)
- Multiply by y:
8. What is the difference between logarithmic differentiation and implicit differentiation?
Logarithmic differentiation is a special case of implicit differentiation where you first apply a logarithm before differentiating. The difference is:
- Implicit differentiation: Used when y is not explicitly solved in terms of x
- Logarithmic differentiation: Used to simplify complex expressions using logarithms before applying implicit differentiation
9. What are common mistakes in logarithmic differentiation?
Common mistakes in logarithmic differentiation include forgetting the chain rule and not multiplying back by y at the end. Key errors to avoid:
- Forgetting that d/dx(ln y) = (1/y) dy/dx
- Not simplifying logarithms correctly
- Forgetting to substitute back y = f(x)
- Ignoring domain restrictions (e.g., ln requires positive values)
10. Is logarithmic differentiation only for natural logarithms?
Logarithmic differentiation typically uses the natural logarithm (ln) because its derivative is simple: d/dx(ln x) = 1/x. While other bases like log base 10 can be used, they introduce extra constants due to change-of-base formulas. Therefore, ln is preferred for cleaner and simpler derivatives in calculus.
