Evaluate $\mathrm{f}'(x)$ given: - $\mathrm{f}(x)=4x^7$
- $\mathrm{f}(x)=5$
Differentiate $$\mathrm{f}(x)=3x+x^2$$ from first principles.
$\displaystyle\lim_{h\to0}\dfrac{3(x+h)+(x+h)^2-3x-x^2}{h}$ <br> $\displaystyle\lim_{h\to0}\dfrac{3h + 2xh + h^2}{h} = 3 + 2x$
Find the gradient of the curve $$y=x^2-5x+10$$ at the two points where it meets the line $y=4$.
$-1, 1$
$$\mathrm{f}(x)=x^2-2x-8$$ Find where the graph $y=\mathrm{f}'(x)$ crosses the $x$ axis.
$(1,0)$
Differentiate $$\mathrm{f}(x)=2x^3$$ from first principles.
$\displaystyle\lim_{h\to0}\dfrac{2(x+h)^3-2x^3}{h}$ <br> $\displaystyle\lim_{h\to0}\dfrac{6x^2h+3xh^2+h^3}{h} = 6x^2$
Find $\mathrm{f}'(x)$ given: - $\mathrm{f}(x)=\dfrac{3}{x}$
- $\mathrm{f}(x)=\dfrac{1}{3x}$
- $-\dfrac{3}{x^2}$
- $-\dfrac{1}{3x^2}$
Find $\dfrac{\mathrm{d}y}{\mathrm{d}x}$ given: - $y=2x^{-2}+\sqrt{x}$
- $y=\dfrac{7+x^3}{2x}$
- $-4x^{-3}+\dfrac{1}{2\sqrt{x}}$
- $-\dfrac{7}{2x^2} + x$
Find the gradient of the following curves at the specified points: - $y=\dfrac{1}{\sqrt{x}}$ at $(0.25,2)$
- $y=\dfrac{2x-6}{x^2}$ at $(3,0)$
Differentiate $$\mathrm{f}(x)=\dfrac{3}{x}$$ from first principles.
$\displaystyle\lim_{h\to0}\dfrac{\frac{3}{x+h}-\frac{3}{x}}{h}$ <br> $\displaystyle\lim_{h\to0}\dfrac{3x-3x-3h}{xh(x+h)} = -\dfrac{3}{x^2}$
Given $$\mathrm{f}(x)=\dfrac{a}{\sqrt{x}}+2x$$ and that $\mathrm{f}'(2)=5$, find $a$
$-12\sqrt{2}$
Given $\mathrm{f}(x)=(2-x)^9$, show that $$\mathrm{f}'(x)\approx9216x-2304$$ if $x$ is small enough such that the $x^2$ and higher powers of $x$ terms can be ignored.
See video.
Find the equations of the tangents to the following curves at the specified points: - $y=x^2-7x+10$ at $(2,0)$
- $y=4\sqrt{x}$ at $(9,12)$
- $y=-3x+6$
- $y=\dfrac{2}{3}x+6$
Find the equations of the tangents to the following curves at the specified points: - $y=\dfrac{2x-1}{x}$ at $(1,1)$
- $y=x^2-\dfrac{8}{\sqrt{x}}$ at $(4,12)$
- $y=x$
- $y=\dfrac{17}{2}x-22$
The normals to the curve $y=x(1+x^2)$ at the points $(0,0)$ and $(1,2)$ meet. Find the coordinates of their intersection.
$(-3,3)$
The point $P$ with $x$ coordinate $0.5$ lies on the curve $$y=2x^2$$ The normal to the curve at the point $P$ intersects the curve at another point. Find the coordinates of the other point of intersection.
$\left(-\dfrac{3}{4},\dfrac{9}{8}\right)$
A curve has the equation $$y=x^2+ax+b$$ where $a$ and $b$ are constants. The minimum point has coordinates $(-2,5)$. Find the values of $a$ and $b$.
$a=4$, $b=9$
Find the coordinates of the stationary points of the curve $$y=2+6x^2-x^3$$ Determine whether each stationary point is a maximum or minimum point.
$(0,2)$ min, $(4,34)$ max
Find the set of values of $x$ for which $$x^3+4x^2-3x+7$$ is increasing.
$\{x: x < -3\} \cup \{x: x > \frac{1}{3}\}$
The curve $$y=x^4-6x^2+7x+2$$ has two points of inflection. Find their coordinates.
$(-1,-10)$ and $(1,4)$
Find values of $x$ corresponding to the convex and concave sections of the curve $$y=x^3+2x^2+4$$
Convex: $x > -\frac{2}{3}$, Concave: $x < -\frac{2}{3}$
The curve $$y=x^3+bx^2+cx+5$$ has a single stationary point. Find $c$ in terms of $b$.
$\dfrac{b^2}{3}$
The curve $$y=4x+ax^2-x^3$$ is concave for $x > 1$. Find $a$.
$3$
A curve has equation $$y=x^3+ax^2-15x+b$$ where $a$ and $b$ are constants, and is stationary at the point $(-1,12)$. Find the coordinates of the other stationary point of the curve.
$(5,-96)$
Amy says that $$y = x^4 + x$$ has a point of inflection at $(0,0)$. Is Amy correct?
No. Even though the second derivative is zero, it does not change sign around the point (it is always positive) so the curve is never concave.