@?differentiation

See Also [[Maths - Differentiation]]S [[Maths - Product Rule]]S [[Maths - Chain Rule]]S Flashcards What is the Quotient Rule used for?? Finding the derivative of two things being divided. $$\frac{d}{dx}\left(\frac{u}{v}\right)$$ What is this equal to?? $$ \frac{v \cdot \frac{du}{dx} - u \cdot \frac{dv}{dx}}{v^2} $$ What is important you remember about the product rule?? The order. Where does the $v^2$ bit come from in the product rule?? It’s actually the product rule with $v^{-1}$, which goes to $\frac{1}{v^2}$

How could you imagine any function $f(x)$ could be written as a polynomial?? $$ f(x) = a_0 + a_1 x + a_2 x^2 + a_3 x^3 + … + a_r x^r $$ $f(x) = a_0 + a_1 x + a_2 x^2 + a_3 x^3 + … + a_r x^r$ If you wanted to work out $a_0$, what could you set $x$ equal to?? $$ 0 $$ $f(x) = a_0 + a_1 x + a_2 x^2 + a_3 x^3 + … + a_r x^r$ What happens if you substitute in $x = 0$?

See Also [[Maths - Differentiation]]S [[Maths - Chain Rule]]S Flashcards If $y = uv$, what is $\frac{dy}{dx}$?? $$ u\frac{dv}{dx} + v\frac{du}{dx} $$ If $f(x) = g(x)h(x)$, what is $f'(x)$?? $$ g(x)h'(x) + h(x)g'(x) $$ How would you explain the product rule in English?? To find the derivative of two things multiplied together, add the products of the pairs of each function and its opposite’s derivative. This diagram represents $h(x) = f(x)g(x)$.

See Also [[Maths - Differentiation]]S https://betterexplained.com/articles/derivatives-product-power-chain/ Flashcards What is the chain rule?? $$ \frac{dy}{dx} = \frac{dy}{du} \times \frac{du}{dx} $$ What new variable do you introduce when using the chain rule?? $$ u $$ When do you apply the chain rule?? When you have composite functions. $$y = (3x + 4)^5$$ What substitution would you make in order to differentiate?? $$ u = 3x + 4 \\ y = u^5 $$ $$y = u^5$$ What is $\frac{dy}{du}$?

See Also [[Maths - Differentiation]]S Flashcards Visually, if $y = f(x)$ is a maximum or minimum, what does the curve $y = f'(x)$ do?? Cuts the $x$-axis. Visually, if $y = f(x)$ is a point of inflection, what does the curve $y = f'(x)$ do?? Touches the $x$-axis. Visually, if $y = f(x)$ has a positive gradient, where is the curve $y = f'(x)$?? Above the $x$-axis. Visually, if $y = f(x)$ has a negative gradient, where is the curve $y = f'(x)$?

What would the differential be called for $A = \pi r^2$?? $$ \frac{dA}{dr} $$ $$A = \pi r^2 \ \frac{dA}{dr}$$ How would you describe the differential?? The rate of change of area with respect to radius. Can you differentiate $V = \frac{4}{3} \pi r^3$?? $$ \frac{dV}{dr} = 4\pi r^2 $$ $$V = \frac{4}{3} \pi r^3 \ \frac{dV}{dr} = 4\pi r^2$$ How could you explain “the rate of change of volume with respect to radius”?? How much additional volume you gain for a small change in the radius.