Calculate the derivative of f(x) = 5x^5. (2016)

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Question: Calculate the derivative of f(x) = 5x^5. (2016)

Options:

Correct Answer: 25x^4

Exam Year: 2016

Solution:

The derivative f\'(x) = d/dx(5x^5) = 25x^4.

Calculate the derivative of f(x) = 5x^5. (2016)

Calculate the derivative of f(x) = 5x^5. (2016)

Step 1: Identify the function you want to differentiate. In this case, f(x) = 5x^5.
Step 2: Recall the power rule for derivatives. The power rule states that if you have a term in the form of ax^n, the derivative is a * n * x^(n-1).
Step 3: Apply the power rule to the term 5x^5. Here, a = 5 and n = 5.
Step 4: Multiply the coefficient (5) by the exponent (5). This gives you 5 * 5 = 25.
Step 5: Decrease the exponent by 1. The original exponent was 5, so 5 - 1 = 4.
Step 6: Combine the results from Step 4 and Step 5. The derivative is 25x^4.
Step 7: Write the final answer as f'(x) = 25x^4.

Power Rule – The derivative of x^n is n*x^(n-1), applied here with a constant multiplier.
Constant Multiple Rule – When differentiating a constant multiplied by a function, the constant remains and only the function is differentiated.