Determine whether each of the following statements is true or false, and explain why: The derivative of 10x is x10x-1.
The Correct Answer and Explanation is:
The statement “The derivative of 10x is x10x-1” is false.
To understand why, let’s break it down:
- Derivative of a constant multiplied by a variable: The expression “10x” is a constant multiplied by the variable xxx, where 10 is a constant. The rule for differentiating such an expression is: ddx(c⋅xn)=c⋅n⋅xn−1\frac{d}{dx}(c \cdot x^n) = c \cdot n \cdot x^{n-1}dxd(c⋅xn)=c⋅n⋅xn−1 where ccc is a constant, and nnn is the exponent of xxx. In this case, the exponent of xxx is 1 (since 10x=10⋅x110x = 10 \cdot x^110x=10⋅x1).
- Applying the power rule: According to the power rule of differentiation, the derivative of x1x^1×1 is simply 1. So, applying this to 10x10x10x, we get: ddx(10x)=10⋅1=10\frac{d}{dx}(10x) = 10 \cdot 1 = 10dxd(10x)=10⋅1=10
Thus, the derivative of 10x10x10x is 10, not x10x−1x^{10x-1}x10x−1.
The expression “x10x-1” is not a valid representation of any simple derivative or algebraic simplification. It appears to be a misinterpretation or typographical error. Instead, the derivative of 10x10x10x is simply a constant, 10.
To summarize:
- The derivative of 10x10x10x is 10, not x10x−1x^{10x-1}x10x−1
