Question

Find f'(x)
a)f(x)=e^x+x^e
b)f(x)=3x^e-2e^x
c)f(x)=xx^e
f(x)=ee^x

Answer 1

$a)\ f(x)=e^x+x^e\\ \\ f'(x)= \frac{d}{dx}(e^x+x^e) \\ \\ \Rightarrow f'(x)= \frac{d}{dx}e^x+\frac{d}{dx}x^e \\ \\ \Rightarrow f'(x)=e^x+ex^{e-1}$


$b)\ f(x)=3x^e-2e^x\\ \\f'(x)= \frac{d}{dx}( 3x^e-2e^x)\\ \\ \Rightarrow f'(x)= \frac{d}{dx}(3x^e)-\frac{d}{dx}(2e^x)\\ \\ \Rightarrow f'(x)= 3\frac{d}{dx}(x^e)-2\frac{d}{dx}(e^x)\\ \\ \Rightarrow f'(x)=3ex^{e-1}-2e^x$


$c)\ f(x)=xx^e=x^{e+1}\\ \\ f'(x)= \frac{d}{dx}(x^{e+1}) \\ \\ \Rightarrow f'(x)=(e+1)x^e$


$d)\ f(x)=ee^x=e^{x+1}\\ \\ f'(x)= \frac{d}{dx}(e^{x+1})\\ \\ \Rightarrow f'(x)=e^{x+1}\frac{d}{dx}(x+1)\\ \\ \Rightarrow f'(x)=e^{x+1} \times 1=e^{x+1}$