Find $\dfrac{\mathrm{d}}{\mathrm{d}x} (x^6 + 2x^3)$
$6x^5 + 6x^2$
Find $\dfrac{\mathrm{d}}{\mathrm{d}x} (3x^4 - 2x)$
$12x^3 - 2$
Find $\dfrac{\mathrm{d}}{\mathrm{d}x}(5x^7 - 4x^2 + 9)$
$35x^6 - 8x$
Find $\dfrac{\mathrm{d}}{\mathrm{d}x} (3 + \sqrt{x})$
$\dfrac{1}{2\sqrt{x}}$
Find $\dfrac{\mathrm{d}}{\mathrm{d}x} \left(\dfrac{1}{2x} + \dfrac{3}{x^2}\right)$
$-\dfrac{1}{2}x^{-2} - 6x^{-3}$
Find $\dfrac{\mathrm{d}}{\mathrm{d}x} \left(\sqrt[3]{x} - \dfrac{1}{2\sqrt{x}}\right)$
$\dfrac{1}{3}x^{-\frac{2}{3}} + \dfrac{1}{4}x^{-\frac{3}{2}}$
Find $\dfrac{\mathrm{d}}{\mathrm{d}x}\left(x^{\frac{1}{3}} + 4x^{\frac{2}{5}}\right)$
$\dfrac{1}{3}x^{-\frac{2}{3}} + \dfrac{8}{5}x^{-\frac{3}{5}}$
Find $\dfrac{\mathrm{d}}{\mathrm{d}x}(\dfrac{3}{x^3} - 2x^{-1})$
$-9x^{-4} + 2x^{-2}$
Find $\dfrac{\mathrm{d}}{\mathrm{d}x}(x+1)(x-3)$
$2x - 2$
Find $\dfrac{\mathrm{d}}{\mathrm{d}x}(2x-5)(1+3x)$
$12x - 13$
Find $\dfrac{\mathrm{d}}{\mathrm{d}x}\left(\dfrac{x^2 + 1}{x}\right)$
$1 - x^{-2}$
Find $\dfrac{\mathrm{d}}{\mathrm{d}x}\left(\dfrac{1 - 3x}{3x}\right)$
$-\dfrac{1}{3}x^{-2}$
Given $f(x) = 2x^{\frac{3}{4}} - \dfrac{1}{2}$, find $f'(x)$
$\dfrac{3}{2}x^{-\frac{1}{4}}$
Given $y = (\sqrt{x})^3$, find $\dfrac{\mathrm{d}y}{\mathrm{d}x}$
$\dfrac{3}{2}x^{\frac{1}{2}}$
Given $y = x^2\sqrt{x}$, find $\dfrac{\mathrm{d}y}{\mathrm{d}x}$
$\dfrac{5}{2}x^{\frac{3}{2}}$
Given $a = b(b + 3)$, find $\dfrac{\mathrm{d}a}{\mathrm{d}b}$
$2b + 3$
Given $x = (y + 3)(2y - 1)$, find $\dfrac{\mathrm{d}x}{\mathrm{d}y}$
$4y + 5$
Given $p = \dfrac{5 + \sqrt{q}}{q}$, find $\dfrac{\mathrm{d}p}{\mathrm{d}q}$
$-5q^{-2} - \dfrac{1}{2}q^{-\frac{3}{2}}$
For $y = 8 - \dfrac{2}{x}$, find $\dfrac{\mathrm{d}y}{\mathrm{d}x}$ and $\dfrac{\mathrm{d}^2y}{\mathrm{d}x^2}$
$\dfrac{\mathrm{d}y}{\mathrm{d}x} = 2x^{-2}, \quad \dfrac{\mathrm{d}^2y}{\mathrm{d}x^2} = -4x^{-3}$
For $y = \dfrac{3x^5 - 3}{x^2}$, find $\dfrac{\mathrm{d}y}{\mathrm{d}x}$ and $\dfrac{\mathrm{d}^2y}{\mathrm{d}x^2}$
$\dfrac{\mathrm{d}y}{\mathrm{d}x} = 9x^2 + 6x^{-3}, \quad \dfrac{\mathrm{d}^2y}{\mathrm{d}x^2} = 18x - 18x^{-4}$
Differentiate $x^2 + 3x$ from first principles.
$$\lim_{h\to 0}\frac{(x+h)^2+3(x+h)-(x^2+3x)}{h}$$ $$\lim_{h\to 0}\frac{2xh+h^2+3h}{h} = 2x + 3$$
Differentiate $2x^3 - 5$ from first principles.
$$\lim_{h\to 0}\frac{2(x+h)^3 - 5 - (2x^3 - 5)}{h}$$ $$\lim_{h\to 0}\frac{6x^2h + 6xh^2 + 2h^3}{h} = 6x^2$$
Differentiate $4x - 2x^2$ from first principles.
$$\lim_{h\to 0}\frac{4(x+h) - 2(x+h)^2 - (4x - 2x^2)}{h}$$ $$\lim_{h\to 0}\frac{4h - 4xh - 2h^2}{h} = 4 - 4x$$
Differentiate $5x^3 + 2x^2$ from first principles.
$$\lim_{h\to 0}\frac{5(x+h)^3 + 2(x+h)^2 - (5x^3 + 2x^2)}{h}$$ $$\lim_{h\to 0}\frac{15x^2h + 15xh^2 + 5h^3 + 4xh + 2h^2}{h} = 15x^2 + 4x$$