General derivative formula

$$ 1. \ \frac {d}{dx} (c)=0, \ where \ c \ is \ any \ constant. $$

$$ 2. \ \frac {d}{dx} x^n = nx^{n-1} $$

$$ 3. \ \frac {d}{dx} x=1 $$

$$ 4. \ \frac {d}{dx} [f(x)]^n = n[f(x)]^{n-1} \frac {d}{dx} f(x) $$

$$ 5. \ \frac {d}{dx} \sqrt{x} = \frac {1}{2 \sqrt{x}} $$

$$ 6. \ \frac {d}{dx} \sqrt{f(x)} = \frac {1}{2 \sqrt{f(x)}} \frac {d}{dx} f(x) = \frac {1}{2 \sqrt{f(x)}} f'(x) $$

$$ 7. \ \frac {d}{dx} c \cdot f(x)= c \frac {d}{dx} f(x) = c \cdot f'(x) $$

$$ 8. \ \frac {d}{dx} [f(x) \pm g(x)] = \frac {d}{dx} f(x) \pm \frac {d}{dx} g(x) =f'(x) \pm g'(x) $$

$$ 9. \ \frac {d}{dx} [f(x) \cdot g(x)] = f(x) \frac {d}{dx} g(x) + g(x) \frac {d}{dx} f(x) $$

$$ 10. \ \frac {d}{dx} \left[ \frac {f(x)}{g(x)} \right] = \frac {g(x) \frac {d}{dx} f(x) - f(x) \frac {d}{dx} g(x)}{[g(x)]^2} $$

Example:

Differentiate x⁵ with respect to x.

Solution:

Given, y = x⁵

On differentiating w.r.t we get;

$$ \frac {dy}{dx} = \frac {d}{dx} (x)^5 $$

$$ y’ = 5x^{5-1} = 5x^4 $$

$$ \therefore \frac {d}{dx} (x)^5 = 5x^4 $$