Applying rules of differentiation
Using the Chain Rule
Differentiating \(y = \cos^2\theta\) using the chain rule gives:
\begin{eqnarray*}
y = z^2 && z = \cos\theta\\
\frac{\mathrm{d}y}{\mathrm{d}z} = 2z &&\frac{\mathrm{d}z}{\mathrm{d}\theta} = -\sin \theta\\\\
\frac{\mathrm{d}y}{\mathrm{d}\theta} &=& \frac{\mathrm{d} y}{\mathrm{d} z} \times \frac{\mathrm{d} z}{\mathrm{d}\theta}\\
\frac{\mathrm{d} y}{\mathrm{d}\theta} &=& 2z\times (-\sin\theta) \\
&=& -2\cos\theta\sin\theta\\
&=& -\sin 2\theta
\end{eqnarray*}
To do
- Differentiation using the Chain Rule worksheet (sigma Mathematics and Statistics Support Coventry University)
- Chain Rule: Further Exercises worksheet (sigma Mathematics and Statistics Support Coventry University)
- Chain rule capstone activity (Khan Academy)
More info
- Differentiation Rules (Study Support, USQ Library)
- The chain rule quick reference (mathcentre)
- Chain rule introduction video (Khan Academy)