Derivative a^x
The derivative of a function is a mathematical tool that allows us to measure how the function changes as its input values change
The derivative of a function is a mathematical tool that allows us to measure how the function changes as its input values change. When we want to find the derivative of a function, we are essentially asking the question “how fast is the function changing at a particular point?”
Now, let’s find the derivative of the function f(x) = a^x, where “a” is a constant.
To find the derivative of this function, we can use a powerful mathematical concept called the logarithmic differentiation. Here’s how we can do it:
Step 1: Take the natural logarithm (ln) of both sides of the function: ln(f(x)) = ln(a^x).
Step 2: Apply the logarithmic properties: ln(f(x)) = x * ln(a).
Step 3: Differentiate both sides of the equation using the chain rule. The derivative of ln(f(x)) with respect to x is 1/f(x) * f'(x), and the derivative of x * ln(a) with respect to x is ln(a). So, we have:
1/f(x) * f'(x) = ln(a).
Step 4: Multiply both sides of the equation by f(x): f'(x) = f(x) * ln(a).
Step 5: Substitute f(x) with the original function a^x: f'(x) = a^x * ln(a).
Therefore, the derivative of the function f(x) = a^x is f'(x) = a^x * ln(a).
To summarize, the derivative of the function a^x is a^x multiplied by the natural logarithm of the base “a”.
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