Answer :
The derivative of f (x) = 12x5- 4x4 is option d. f ′ (x) = 60x⁴ -16x³.
The problem requires us to compute the derivative of the function
f (x) = 12x⁵ - 4x⁴.
Calculating the derivative of a function means determining how the function changes at any given point. Here, we use the power rule for differentiation, which is the derivative of xⁿ, where n is any real number, is
n*xⁿ⁻¹.
Step 1:
Let's start with the first term of our function, which is 12x⁵.
According to the power rule, the derivative of xⁿ (x to the power of n) is n*xⁿ⁻¹ where n is any real number. Now applying the power rule to our term:
The power or exponent here is 5, so for our first term, 12x⁵, the derivative will be
5*(12)x⁴.
This simplifies to 60x⁴
Step 2:
Now let's move on to our second term -4x⁴. Again, applying the power rule:
The power or exponent in this term is 4, so for our second term -4x⁴, the derivative will be
4*(-4)x³
This simplifies to -16x³
Step 3:
Now we sum up the derivatives of both terms to get the derivative of the complete function.
So, the derivative of
f (x) = 12x⁵ - 4x⁴ will be
f ′ (x) = 60x⁴ -16x³.
Therefore, the correct option is d. f ′ (x) = 60x⁴ -16x³.
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