The differential of $x^n$, $n \in \mathbb{R}$ is $nx^{n-1}$

If $y = x^2 + 3x$, find $\frac{dy}{dx}$ $ \frac{dy}{dx} = 2x + 3x^0 \

$

$\frac{dy}{dx} = 2x + 3$

e.g. If $y = x^2 + 6x^2 + 5x^4$, find the 1st derivative.

$\frac{dy}{dx} = 2x + 18x + 20x^3$

Find the gradient of $f(x) = 10x^2 + \sqrt{x}$ when $x = 4$.

Must first make all terms powers of $x$ before differentiating.

$f(x) = 10x^2 + x^{\frac{1}{2}}$

$ f'(x) = 20x + \frac{1}{2}x^{-\frac{1}{2}} \

$

$f'(4) = 20(4) + \frac{1}{2}(4)^{-\frac{1}{2}} = \frac{321}{4}$

Note: The expression we differentiate cannot have $x$'s on the denominator.

Find the differential of $f(x)=\dfrac {x^{\frac {1}{2}}+x^{-\frac {1}{2}}}{x^1}$ when $x = 16$ , showing detailed reasoning.

$f(x) = x^{-\frac{1}{2}} + x^{-\frac{3}{2}}$

$f'(x) = \frac{-1}{2}x^{-\frac{-3}{2}} - \frac{3}{2}x^{-\frac{-5}{2}}$

When $x = 16$:

$f'(16) = \frac{-1}{2}(16)^{-\frac{-3}{2}} - \frac{3}{2}(16)^{-\frac{5}{2}} = \frac{-19}{2048}$

Find the rate of change with respect to $x$ of $f(x) = \frac{x^2 + x^3}{^3\sqrt{x}}$ when $x = 1$, showing detailed reasoning. $f(x) = \frac{x^2}{x^{\frac{1}{3}}} + \frac{x^3}{x^{\frac{1}{3}}} = x^{\frac{5}{3}} + x^{\frac{8}{3}}$

$\Rightarrow f'(x) = \frac{5}{3}x^{\frac{2}{3}} + \frac{8}{3}x^{\frac{5}{3}}$

$\Rightarrow f'(1) = \frac{5}{3}(1)^{\frac{2}{3}} + \frac{8}{3}(1)^{\frac{5}{3}}= \frac{13}{3}$

Find $\frac{dy}{dx}$ when:

(a) $y = \frac{2}{5x} = \frac{2}{5}x^{-1}$

$\frac{dy}{dx} = -\frac{2}{5}x^{-2}$

(b) $y = \frac{7}{5^4\sqrt{x}} = \frac{7}{5}x^{-\frac{1}{4}}$

$\frac{dy}{dx} = -\frac{7}{10}x^{-\frac{5}{4}}$