Differentiation — common mistakes

In connected rates of change problems, always start by writing down what you know (e.g., $\frac{dV}{dt} = 5$) and what you want to find (e.g., $\frac{dr}{dt}$). Then, write down the chain rule relationship that connects these rates, such as $\frac{dr}{dt} = \frac{dr}{dV} \times \frac{dV}{dt}$. This helps structure your solution.

Use the product rule when you are differentiating a function that is one function MULTIPLIED by another, like $f(x) = x^2 e^x$. Use the chain rule when you have a function INSIDE another function (a composite function), like $f(x) = e^{(x^2)}$. Sometimes you need both, for example in $f(x) = x^2 e^{3x}$ you use the product rule overall, but need the chain rule to differentiate the $e^{3x}$ part.

Sometimes, yes. You can often rewrite a quotient as a product and use the product rule. For example, $y = \frac{\ln x}{x^2}$ can be written as $y = (\ln x)(x^{-2})$. Differentiating this with the product rule can sometimes feel simpler and lead to fewer sign errors. However, you must be comfortable with both methods.

The second derivative test is usually much faster than testing points on either side of the stationary point. It's only inconclusive in specific cases (like $y=x^3$ or $y=x^4$ at $x=0$). For most exam questions, it works perfectly. You just need to remember the backup method (testing the sign of $f'(x)$) for when $f''(x)=0$.

Look at the rate you are given and the rate you need to find. For example, if you are given $\frac{dV}{dt}$ and need to find $\frac{dh}{dt}$, your chain rule will be $\frac{dh}{dt} = \frac{dh}{dV} \times \frac{dV}{dt}$. This tells you that you need a relationship between $h$ and $V$. Therefore, you should write your volume formula $V$ in terms of $h$ only, eliminating other variables like the radius $r$.