All Derivatives Formulas in Calculus – Class 12

📘 Derivative Formulas

🔹 1. Basic Rules

\[ \frac{d}{dx}(c) = 0 \quad \text{(c = constant)} \] \[ \frac{d}{dx}(x) = 1 \] \[ \frac{d}{dx}(x^n) = n x^{n-1} \quad \text{(for any real } n) \]

🔹 2. Trigonometric Functions

\[ \frac{d}{dx}(\sin x) = \cos x \] \[ \frac{d}{dx}(\cos x) = -\sin x \] \[ \frac{d}{dx}(\tan x) = \sec^2 x \] \[ \frac{d}{dx}(\cot x) = -cosec^2 x \] \[ \frac{d}{dx}(\sec x) = \sec x \tan x \] \[ \frac{d}{dx}(cosec x) = -cosec x \cot x \]

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🔹 3. Exponential and Logarithmic Functions

\[ \frac{d}{dx}(e^x) = e^x \] \[ \frac{d}{dx}(a^x) = a^x \log a \] \[ \frac{d}{dx}(\log x) = \frac{1}{x} \] \[ \frac{d}{dx}(\log_a x) = \frac{1}{x \ln a} \]

🔹 4. Inverse Trigonometric Functions

\[ \frac{d}{dx}(\sin^{-1} x) = \frac{1}{\sqrt{1 – x^2}} \] \[ \frac{d}{dx}(\cos^{-1} x) = -\frac{1}{\sqrt{1 – x^2}} \] \[ \frac{d}{dx}(\tan^{-1} x) = \frac{1}{1 + x^2} \] \[ \frac{d}{dx}(\cot^{-1} x) = -\frac{1}{1 + x^2} \] \[ \frac{d}{dx}(\sec^{-1} x) = \frac{1}{x \sqrt{x^2 – 1}} \] \[ \frac{d}{dx}(\csc^{-1} x) = -\frac{1}{x \sqrt{x^2 – 1}} \]

🔹 5. Rules of Differentiation

\[ \frac{d}{dx}[f(x) \pm g(x)] = f'(x) \pm g'(x) \] \[ \frac{d}{dx}[f(x)g(x)] = f'(x)g(x) + f(x)g'(x) \] \[ \frac{d}{dx}\left[\frac{f(x)}{g(x)}\right] = \frac{f'(x)g(x) – f(x)g'(x)}{[g(x)]^2} \] \[ \text{If } y = f(g(x)), \quad \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} \]\[ \frac{d}{dx}[u(x) \cdot v(x)] = u'(x) \cdot v(x) + u(x) \cdot v'(x) \]

🔹 6. Logarithmic Differentiation Example

\[ y = x^x \Rightarrow \log y = x \log x \] \[ \frac{d}{dx}(\log y) = \frac{1}{y} \cdot \frac{dy}{dx} = \log x + 1 \Rightarrow \frac{dy}{dx} = y (\log x + 1) = x^x (\log x + 1) \]

🔹 7. Second Order Derivative

\[ \frac{d^2y}{dx^2} = \frac{d}{dx}\left(\frac{dy}{dx}\right) \]