Advanced differentiation problems : Exercises
Introduction
To solve the following problems you will need to use everything you know about the
differential operator. It will be particularly important to remember that the
differential operator has the distributative property over addition, that the chain
rule allows us to evaluate the differential of the function $f[u(x)]$ with respect
to $x$ using:
$$
\frac{\textrm{d}f}{\textrm{d}x} = \frac{\textrm{d}f}{\textrm{d}u} \frac{\textrm{d}u}{\textrm{d}x}
$$
Last of all you will need to remember that the product rule can be used to calculate
the differentials of products of functions such as $f(x) = u(x)v(x)$. The product
rule states in this case that:
$$
\frac{\textrm{d}f}{\textrm{d}x} = u \frac{\textrm{d}v}{\textrm{d}x} + v\frac{\textrm{d}u}{\textrm{d}x}
$$
Example problems
Click on the problems to reveal the solution
Problems for you to try
| $y=x^3 \ln x - \frac{x^3}{3}$ |
$y = 2t\sin t - (t^2 - 2\cos t)$ |
$f(t) = \sin(t)\sin(t+\phi)$ |
$y = \frac{1+ \sqrt{z}}{1 - \sqrt{z} }$ |
| $y = 5e^{-ax^2}$ |
$y = \sqrt{a^2 - x^2}$ |
$y = (3-2\sin(x))^5$ |
Problems for you to try
Find the derivative of the following function with respect to $y$:
$$
f(y) = \frac{\int_a^b v(x) \exp\left( - \frac{\phi(y,x)}{2\sigma^2} \right) \textrm{d}x }{ \int_a^b \exp\left( - \frac{\phi(y,x)}{2\sigma^2} \right) \textrm{d}x }
$$
Hint: $\frac{\partial \phi(y,x)}{\partial y} = \phi'(y,x)$
