medium difficulty7 min

Derivatives of Inverse Trigonometric, Exponential, and Logarithmic Functions

This concept covers standard derivatives of inverse trigonometric functions, exponential functions, and logarithmic functions, along with their domain conditions.

In one line: Inverse sine is positive, inverse cosine is negative, log gives denominator, exponential repeats itself.

Practice This Concept

Understand it

These derivatives are used directly and also inside chain rule questions. Domain matters: log x is defined for x > 0, and inverse trigonometric derivative formulas such as sin⁻¹ x require expressions where the denominator remains meaningful.

Example

If y = log(1 + x²), then dy/dx = 1/(1 + x²) × 2x = 2x/(1 + x²).

Formula and rule

d/dx(sin⁻¹ x) = 1/√(1 - x²), d/dx(cos⁻¹ x) = -1/√(1 - x²), d/dx(tan⁻¹ x) = 1/(1 + x²), d/dx(e^x) = e^x, d/dx(a^x) = a^x log a, d/dx(log x) = 1/x for x > 0.

With u(x), multiply each by u'(x).

See it worked out

Follow the example line by line, then use the same method on new questions.

Worked example

Differentiate y = tan⁻¹(3x) + log(x² + 1).

1For tan⁻¹(3x), derivative = 3/(1 + 9x²).
2For log(x² + 1), derivative = 2x/(x² + 1).
3Hence dy/dx = 3/(1 + 9x²) + 2x/(x² + 1).

Method you can reuse

1Identify whether the function is inverse trigonometric, exponential, or logarithmic.
2Check the inner expression.
3Apply the standard formula.
4Multiply by the derivative of the inner expression.
5Mention domain where logarithmic or square-root denominator conditions are relevant.

Visual tip

Graphs are not necessary for standard derivative calculation; domain restrictions should be stated algebraically.

Check yourself

Answer the question in your head before opening the answer.

Try it yourself

Differentiate y = e^(4x - 1).

Think first, then show answer

dy/dx = e^(4x - 1) × 4 = 4e^(4x - 1).

Common wrong answer

Writing d/dx[cos⁻¹ x] = 1/√(1 - x²) and missing the negative sign.

Don't mix this up

Students often write the correct outer derivative but ignore the derivative of the inside expression, especially in sin⁻¹(2x) or log(3x + 1).

Exam tip

For inverse trigonometric functions, write the formula first, then substitute the inner expression and multiply by its derivative.

Write it in the exam

Each point is worth about one mark. Write point 1 for a 1-mark question and keep adding points as the marks go up.

How CBSE asks it

Usually asked as direct differentiation or mixed with chain rule, product rule, quotient rule, and logarithmic differentiation.

1
Enough for 1 mark
This concept covers standard derivatives of inverse trigonometric functions, exponential functions, and logarithmic functions, along with their domain conditions.
2
Add for 2 marks
Example: If y = log(1 + x²), then dy/dx = 1/(1 + x²) × 2x = 2x/(1 + x²).
3
Add for 3 marks
Domain matters: log x is defined for x > 0, and inverse trigonometric derivative formulas such as sin⁻¹ x require expressions where the denominator remains meaningful.

MCQ Quiz

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