find the derivative of ( f(x) ).\n\n( f(x)=4 cdot 3^{x}-4^{x}+5 cdot 7^{x} )\n\n( f^{prime}(x)= )

Answer:

$4\cdot3^x\ln3 - 4^x\ln4+5\cdot7^x\ln7$

Explanation:

Step1: Use derivative rule for $a^x$

The derivative of $a^x$ is $a^x\ln a$.

Step2: Differentiate each term

• For the term $4\cdot3^x$: Using the constant - multiple rule $(cf(x))^\prime = cf^\prime(x)$ and the rule for $a^x$, its derivative is $4\cdot3^x\ln3$.
• For the term $-4^x$: Its derivative is $-4^x\ln4$.
• For the term $5\cdot7^x$: Using the constant - multiple rule and the rule for $a^x$, its derivative is $5\cdot7^x\ln7$.

Step3: Combine the derivatives

Combine the derivatives of each term: $f^\prime(x)=4\cdot3^x\ln3 - 4^x\ln4+5\cdot7^x\ln7$.