Use the product rule to differentiate the following with respect to $x$: $$(x-1)(x+5)$$
$2x + 4$
Use the product rule to differentiate the following with respect to $x$: $$(2-x)(2x+1)$$
$3-4x$
Use the product rule to differentiate the following with respect to $x$: $$(1-2x^2)(3x+4)$$
$-18x^2 - 16x + 3$
Use the product rule to differentiate the following with respect to $x$: $$(x^2 + x + 6)(3x - 2)$$
$9x^2 + 2x + 16$
Given $$y = x(x+2)^3$$ find $\dfrac{\mathrm{d}y}{\mathrm{d}x}$. Give your answer in the form $(ax+b)(x+2)^2$.
$(x+2)^3 + 3x(x+2)^2 = (4x+2)(x+2)^2$
Given $$y = x(x-3)^4$$ find $\dfrac{\mathrm{d}y}{\mathrm{d}x}$. Give your answer in the form $(ax+b)(x-3)^3$.
$(x-3)^4 + 4x(x-3)^3 = (5x-3)(x-3)^3$
Given $$y = x^2(x-1)^4$$ find $\dfrac{\mathrm{d}y}{\mathrm{d}x}$. Give your answer in the form $ax(bx+c)(x-1)^3$.
$2x(x-1)^4 + 4x^2(x-1)^3 = 2x(3x - 1)(x-1)^3$
Given $$y = 2x^3(3+x)^5$$ find $\dfrac{\mathrm{d}y}{\mathrm{d}x}$. Give your answer in the form $ax^2(bx+c)(3+x)^4$.
$6x^2(3+x)^5 + 10x^3(3+x)^4 = 2x^2(3 + 5x)(3+x)^4$
Given $\mathrm{f}(x) = x^3(2x+1)^5$, find $\mathrm{f}'(x)$ in the form $x^2(ax+b)(2x+1)^4$.
$3x^2(2x+1)^5 + 10x^3(2x+1)^4 = x^2(3+10x)(2x+1)^4$
Given $\mathrm{f}(x) = 3x^2(1-2x)^6$, find $\mathrm{f}'(x)$ in the form $ax(bx+c)(1-2x)^5$.
$6x(1-2x)^6 - 36x^2(1-2x)^5 = 6x(1-8x)(1-2x)^5$
Given $\mathrm{f}(x) = x\sqrt{2x+4}$, find $\mathrm{f}'(x)$ in the form $(ax+b)(2x+4)^{-\frac{1}{2}}$.
$(2x+4)^{\frac{1}{2}} + x(2x+4)^{-\frac{1}{2}} = (3x+4)(2x+4)^{-\frac{1}{2}}$
Given $\mathrm{f}(x) = 3x^2\sqrt{1-4x}$, find $\mathrm{f}'(x)$ in the form $ax(bx+c)(1-4x)^{-\frac{1}{2}}$.
$6x(1-4x)^{\frac{1}{2}} - 6x^2(1-4x)^{-\frac{1}{2}} = 6x(1-5x)(1-4x)^{-\frac{1}{2}}$
Find the coordinates of the stationary points on the curve $$y = x(x-2)^3$$
$\left(\dfrac{1}{2},-\dfrac{27}{16}\right)$, $(2,0)$
Find the coordinates of the stationary points on the curve $$y = x^2(2x+3)^4$$
$(-1.5,0)$, $(-0.5,4)$, $(0,0)$
Find the coordinates of the stationary points on the curve $$y = (2-x)(3+x)^4$$
$(-3,0)$, $(1,256)$
Find the coordinates of the stationary points on the curve $$y = (1+2x)(1-3x)^4$$
$\left(-\dfrac{1}{3},\dfrac{16}{3}\right)$, $\left(\dfrac{1}{3},0\right)$
Find the equation of the tangent to the curve $$y = x(2x+4)^4$$ at the point where $x = 0$.
$y = 256x$
Find the equation of the tangent to the curve $$y = \sqrt{x}(x+2)^3$$ at the point where $x = 1$. Give your answer in the form $ax + by + c = 0$.
$81x - 2y - 27 = 0$
Find the equation of the normal to the curve $$y = (x+3)(x-2)^3$$ at the point where $x = 1$. Give your answer in the form $ax + by + c = 0$.
$x + 11y + 43 = 0$
Find the equation of the normal to the curve $$y = (2x-1)(2-x)^3$$ at the point where $x = 1$.
$y = x$
Differentiate the following with respect to $x$: $$\dfrac{x}{x+2}$$
$\dfrac{2}{(x+2)^2}$
Differentiate the following with respect to $x$: $$\dfrac{x}{2-x}$$
$\dfrac{2}{(2-x)^2}$
Differentiate the following with respect to $x$: $$\dfrac{x^2}{1-2x}$$
$\dfrac{2x(1-x)}{(1-2x)^2}$
Differentiate the following with respect to $x$: $$\dfrac{x}{2-x^2}$$
$\dfrac{x^2+2}{(2-x^2)^2}$
Differentiate the following with respect to $x$: $$\dfrac{2x+1}{\sqrt{1-2x}}$$
$\dfrac{3-2x}{(1-2x)^{\frac{3}{2}}}$
Differentiate the following with respect to $x$: $$\dfrac{(1+2x)}{\sqrt{3-x}}$$
$\dfrac{13-2x}{2(3-x)^{\frac{3}{2}}}$
Differentiate the following with respect to $x$: $$\left(\dfrac{x+1}{3-2x}\right)^2$$
$\dfrac{10(x+1)}{(3-2x)^3}$
Differentiate the following with respect to $x$: $$\sqrt{\dfrac{x-2}{2x-3}}$$
$\dfrac{1}{2(2x-3)^{\frac{3}{2}}\sqrt{x-2}}$