A-Level CAIE Mathematics A2 1.7 Differentiation Question Bank

A curve has equation $y=2 x^{\frac{3}{2}}-3 x-4 x^{\frac{1}{2}}+4$. Find the equation of the tangent to the curve at the point (4,0).

Find the gradient of the curve

at the point for which x=0.

It is given that $\mathrm{f}(x)=(2 x-5)^{3}+x$, for $x \in \mathbb{R}$. Show that f is an increasing function.

The equation of a curve is $y=\frac{1+x}{1+2 x}$ for $x>-\frac{1}{2}$. Show that the gradient of the curve is always negative.

The function f is defined by $\mathrm{f}(x)=\frac{1}{3 x+2}+x^{2}$ for x<-1.
Determine whether f is an increasing function, a decreasing function or neither.

A curve has equation

Find the coordinates of the points on the curve at which the gradient is -4 .

A curve has equation $y=3 \ln (2 x+9)-2 \ln x$.

The equation of a curve is such that $\frac{\mathrm{d} y}{\mathrm{~d} x}=12\left(\frac{1}{2} x-1\right)^{-4}$. It is given that the curve passes through the point P(6,4).

A function f is defined by $\mathrm{f}: x \mapsto x^{3}-x^{2}-8 x+5$ for x<a. It is given that f is an increasing function. Find the largest possible value of the constant a.

The volume of a spherical balloon is increasing at a constant rate of $50 \mathrm{~cm}^{3}$ per second. Find the rate of increase of the radius when the radius is 10 cm . [Volume of a sphere $=\frac{4}{3} \pi r^{3}$.]