What is wrong with the following reasoning???
f(x) = 2^x
f'(x) = x[2^(x-1)]
Although n x^(n-1) is the derivative of x^n, you cannot apply similar rules when x is the exponent and the number being raised to a power is a constant.
The derivative of b^x, where b is a constant, is ln(b) * b^x
For a proof, see http://math2.org/math/derivatives/more/b^x.htm
The reasoning provided is incorrect because it applies a different rule for finding the derivative of a function raised to a constant exponent.
The correct rule for finding the derivative of a function f(x) = b^x, where b is a constant, is ln(b) * b^x.
In the given reasoning, the function f(x) = 2^x is being differentiated as if it were x^(2-1) instead of 2^x. The correct application of the derivative rule for b^x yields ln(b) * b^x, which in this case is ln(2) * 2^x.
To understand this concept in detail, you can refer to the provided link: http://math2.org/math/derivatives/more/b^x.htm.
The reasoning is incorrect because it assumes that the derivative of the function f(x) = 2^x is x[2^(x-1)].
The derivative of a function can be found using various rules and formulas, but in this case, the reasoning overlooks the fact that the base of the exponent, 2, is a constant.
When the base of the exponent is a constant, the derivative of b^x, where b is a constant, is given by ln(b) * b^x. This is a well-known property called the exponential function derivative property.
To apply this rule, we can differentiate f(x) = 2^x using ln(b) * b^x. In this case, b = 2, so the derivative of f(x) can be found using the formula ln(2) * 2^x.
It is important to note that the proof of this property can be found in various math resources, such as textbooks or online articles. The link you provided (http://math2.org/math/derivatives/more/b^x.htm) demonstrates the proof of the derivative property for b^x. This link can be a helpful resource to understand the reasoning behind the formula.