Question 28.7.9: Differentiate e^ax cos (b x+c) ....
Differentiate e^{a x} \cos (b x+c) .
Step-by-Step
Using the product rule, we have
\begin{array}{l}\frac{d}{d x}\left\{e^{a x} \cos (b x+c)\right\} \\ \\\quad=e^{a x} \cdot \frac{d}{d x}\{\cos (b x+c)\}+\cos (b x+c) \cdot \frac{d}{d x}\left(e^{a x}\right)\\ \\\quad=e^{a x}\{-\sin (b x+c)\} \cdot \frac{d}{d x}(b x+c)+\cos (b x+c) \cdot e^{a x} \cdot \frac{d}{d x}(a x) \\ \\\text{[using the chain rule]}\\ \\\quad=-b e^{a x} \sin (b x+c)+a e^{a x} \cos (b x+c) \\ \\\quad=e^{a x}[a \cos (b x+c)-b \sin (b x+c)] .\end{array}
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