F(x) = 2^(x + 1) - 2^(x - 1) please help, we need to find the derivative, explain step by step ​

Question

`f(x) = 2^(x + 1) - 2^(x - 1)`

please help, we need to find the derivative, explain step by step ​

Answer 1

To find the derivative of the given function, we can use the power rule and simplify the expression.

Step-by-step explanation:

To find the derivative of the function f(x) = 2^(x + 1) - 2^(x - 1), we can use the power rule. The power rule states that if we have a function of the form g(x) = a^x, the derivative is given by g'(x) = ln(a) * a^x. Applying this rule, we get:

f'(x) = ln(2) * 2^(x + 1) - ln(2) * 2^(x - 1)

Simplifying further, we have:

f'(x) = 2^x * (2 * ln(2) - 1/2 * ln(2))

Answer 2

Note the derivative rule

`\\ \boxed{\star\bf (d)/(dx)(x^n)=nx^(n-1)}`

Let's see

`\\ \rm\Rrightarrow (d)/(dx)(2^(x+1)-2^(x-1))`

`\\ \rm\Rrightarrow (d)/(dx)2^(x+1)-(d)/(dx)2^(x-1)`

`\\ \rm\Rrightarrow (x+1)2^(x+1-1)-(x-1)2^(x-1-1)`

`\\ \rm\Rrightarrow (x+1)2^x-(x-1)2^(x-2)`

`\\ \rm\Rrightarrow (x+1)2^x-(x-1)2^x/2^2`

`\\ \rm\Rrightarrow (x+1)2^x-(x-1)2^x2^(-2)`

`\\ \rm\Rrightarrow 2^x(x+1-(x-1)2^(-2))`

`\\ \rm\Rrightarrow 2^x(x+1-x/4-1/4)`