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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.

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Student-friendly explanation

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.

How to write this in exams

  1. 1

    Start with the exact idea

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

  2. 2

    Then show how to use it

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

  3. 3

    Add one concrete example

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

  4. 4

    Avoid this incomplete answer

    Writing d/dx[cos^-1 x] = 1/sqrt(1 - x^2) and missing the negative sign.

Definition

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

Example

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

Rule to remember

d/dx(sin^-1 x) = 1/sqrt(1 - x^2), d/dx(cos^-1 x) = -1/sqrt(1 - x^2), d/dx(tan^-1 x) = 1/(1 + x^2), 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).

Memory hook

Inverse sine is positive, inverse cosine is negative, log gives denominator, exponential repeats itself.

Examples and method

Worked example

Differentiate y = tan^-1(3x) + log(x^2 + 1). For tan^-1(3x), derivative = 3/(1 + 9x^2). For log(x^2 + 1), derivative = 2x/(x^2 + 1). Hence dy/dx = 3/(1 + 9x^2) + 2x/(x^2 + 1).

Method to apply

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

Diagram support

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

How CBSE asks it

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

Avoid common mistakes

Common confusion

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

Common wrong answer

Writing d/dx[cos^-1 x] = 1/sqrt(1 - x^2) and missing the negative sign.

Exam tip

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

Quick check

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

dy/dx = e^(4x - 1) * 4 = 4e^(4x - 1).

Answer writing and exam use

1-mark answer

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

2-mark answer

This concept covers standard derivatives of inverse trigonometric functions, exponential functions, and logarithmic functions, along with their domain conditions. d/dx(sin^-1 x) = 1/sqrt(1 - x^2), d/dx(cos^-1 x) = -1/sqrt(1 - x^2), d/dx(tan^-1 x) = 1/(1 + x^2), 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). If y = log(1 + x^2), then dy/dx = 1/(1 + x^2) * 2x = 2x/(1 + x^2).

3-mark answer

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. d/dx(sin^-1 x) = 1/sqrt(1 - x^2), d/dx(cos^-1 x) = -1/sqrt(1 - x^2), d/dx(tan^-1 x) = 1/(1 + x^2), 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). Differentiate y = tan^-1(3x) + log(x^2 + 1). For tan^-1(3x), derivative = 3/(1 + 9x^2). For log(x^2 + 1), derivative = 2x/(x^2 + 1). Hence dy/dx = 3/(1 + 9x^2) + 2x/(x^2 + 1). Usually asked as direct differentiation or mixed with chain rule, product rule, quotient rule, and logarithmic differentiation. Writing d/dx[cos^-1 x] = 1/sqrt(1 - x^2) and missing the negative sign.
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