Differentiate and do not simplify.
y= e^(- sq.rt x) + e^3 - 2^(3x-1)
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To differentiate the given function, y = e^(-√x) + e^3 - 2^(3x-1), you can follow these steps:
Step 1: Differentiate each term of the function separately using the chain rule and power rule.
Term 1: y₁ = e^(-√x)
To differentiate this term, apply the chain rule by considering the derivative of the outer function (e^x) and the derivative of the inner function (-√x).
dy₁/dx = d/dx (e^(-√x))
= e^(-√x) * d/dx(-√x)
= e^(-√x) * -1/2 * (1/√x)
= -1/2√x * e^(-√x)
Term 2: y₂ = e^3
The derivative of any constant is always zero.
dy₂/dx = d/dx (e^3)
= 0
Term 3: y₃ = 2^(3x-1)
To differentiate this term, we use the power rule. Take note that the base is 2 and the exponent is a function of x.
dy₃/dx = d/dx (2^(3x-1))
= ln(2) * 2^(3x-1) * d/dx (3x-1)
= ln(2) * 2^(3x-1) * 3
= 3ln(2) * 2^(3x-1)
Step 2: Combine the derivatives of each term to get the final result.
dy/dx = dy₁/dx + dy₂/dx + dy₃/dx
= -1/2√x * e^(-√x) + 0 + 3ln(2) * 2^(3x-1)
Therefore, the derivative of y = e^(-√x) + e^3 - 2^(3x-1) is dy/dx = -1/2√x * e^(-√x) + 3ln(2) * 2^(3x-1).