find the derivative of ( f(x) ).\n\n( f(x)=7 cdot 5^{x}-5 e^{x} )\n\n( f^{prime}(x)= )

Answer:

$7\cdot5^{x}\ln(5)-5e^{x}$

Explanation:

Step1: Differentiate term by term

Use the sum - difference rule of derivatives: $(u - v)'=u'-v'$, where $u = 7\cdot5^{x}$ and $v = 5e^{x}$.

Step2: Differentiate $u = 7\cdot5^{x}$

Use the constant - multiple rule $(cf(x))'=cf'(x)$ and the formula for the derivative of $a^{x}$ which is $(a^{x})'=a^{x}\ln(a)$. For $y = 7\cdot5^{x}$, let $a = 5$ and $c = 7$. Then $u'=7\cdot(5^{x})'=7\cdot5^{x}\ln(5)$.

Step3: Differentiate $v = 5e^{x}$

Use the constant - multiple rule $(cf(x))'=cf'(x)$ and the formula $(e^{x})'=e^{x}$. For $y = 5e^{x}$, let $c = 5$ and $f(x)=e^{x}$. Then $v'=5\cdot(e^{x})'=5e^{x}$.

Step4: Apply the sum - difference rule

Since $f(x)=u - v$, then $f'(x)=u'-v'$. Substitute $u'$ and $v'$: $f'(x)=7\cdot5^{x}\ln(5)-5e^{x}$.