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Given \( f(x) = 12x^5 - 4x^4 \), find its derivative.

A. \( f'(x) = 60x^3 - 16x^2 \)

B. \( f'(x) = 60x^4 - 16x^2 \)

C. \( f'(x) = 60x^5 - 16x^4 \)

D. \( f'(x) = 60x^4 - 16x^3 \)

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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