Answer to Question #154449 in Calculus for ice

Question

g(t)=3t/e^6t

f(x)= e^x sine^x

Answer 1

Here we have $g(t)=\frac{3t}{e^{6t}} \\$

So, differentiating both sides:-

$$g(t)=\frac{3t}{e^{6t}}\\~\\ \Rightarrow g'(t)=\frac{e^{6t}(3)-3t(6e^{6t})}{e^{12t}}\\~\\ \Rightarrow g'(t)=\frac{3e^{6t}-18te^{6t}}{e^{12t}}\\~\\ \Rightarrow g'(t)=\frac{3-18t}{e^{6t}}$$

And also we have $f(x)=e^xsin(e^x)$

So, differentiating both sides:-

$$f(x)=e^xsin(e^x)\\ \Rightarrow f'(x)=sin(e^x)e^x+e^xcos(e^x)e^x\\ \Rightarrow f'(x)=e^xsin(e^x)+e^{2x}cos(e^x)$$

So, we have our derivatives of the two given functions.