Differentiate ax w.r.t. x, where a is a positive constant.
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Step-by-Step Solution
Step 1: Rewrite the expression using base e
To differentiate ax, it's often helpful to rewrite it using the natural exponential function e. We know that any positive number N can be written as elnN. Applying this property, we can express ax as eln(ax).
Step 2: Simplify the exponent
Using the logarithm property ln(MP)=PlnM, we can simplify the exponent. Here, M=a and P=x, so ln(ax) becomes xlna.
Step 3: Differentiate using the chain rule
Now we differentiate exlna with respect to x. We use the chain rule, which states that dxd(ef(x))=ef(x)⋅f′(x). Here, f(x)=xlna.
Step 4: Differentiate the exponent
Since a is a positive constant, lna is also a constant. The derivative of x multiplied by a constant is simply that constant. Therefore, the derivative of xlna with respect to x is lna.
Step 5: Substitute back and simplify
Substitute the derivative of the exponent back into the chain rule result. We know that exlna is equivalent to ax. Therefore, the derivative of ax is axlna.