The Chain Rule | Finding the Derivative of f(x) = a^x

d/dx a^x

To find the derivative of a function, we can use basic differentiation rules

To find the derivative of a function, we can use basic differentiation rules. In the case of the function f(x) = a^x, where a is a constant, we can find its derivative using the chain rule.

The chain rule states that the derivative of a composite function (f(g(x))) is equal to the derivative of the outer function (f'(g(x))) multiplied by the derivative of the inner function (g'(x)).

Now let’s apply the chain rule to find the derivative of f(x) = a^x:

Step 1: Start with the function f(x) = a^x.

Step 2: Take the natural logarithm of both sides of the equation to simplify the function. This step helps us because the derivative of ln(a^x) is simpler.

ln(f(x)) = ln(a^x)

Step 3: Apply the power rule of logarithms to bring down the exponent x in ln(a^x) as a coefficient:

ln(f(x)) = x * ln(a)

Step 4: Now, differentiate both sides of the equation with respect to x:

d/dx [ln(f(x))] = d/dx [x * ln(a)]

Step 5: On the left side of the equation, we are differentiating ln(f(x)) with respect to x. This is equivalent to 1/f(x) multiplied by f'(x) by the chain rule:

1/f(x) * f'(x) = d/dx [x * ln(a)]

Step 6: Substitute f(x) with a^x:

1/a^x * f'(x) = d/dx [x * ln(a)]

Step 7: Simplify further using the properties of logarithms:

1/a^x * f'(x) = ln(a)

Step 8: Now, solve for f'(x), which is the derivative we are interested in:

f'(x) = a^x * ln(a)

So, the derivative of f(x) = a^x is f'(x) = a^x * ln(a).

This result tells us that when we differentiate a function of the form f(x) = a^x, where a is a constant, the derivative is simply the function multiplied by the natural logarithm of the base a.

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