Step-by-Step Guidance for Derivatives Quiz 1 (Calculus)

Q1. Find the derivative of the function: $f(x) = 20x^9 - 4x^7 + 8x^5 - 4x$

Background

Topic: Differentiation (Power Rule)

This question tests your ability to compute the derivative of a polynomial function using the power rule.

Key Terms and Formulas

• Derivative: The rate at which a function changes with respect to its variable.

• Power Rule: For $f(x) = x^n$, the derivative is $f'(x) = n x^{n-1}$.

• Linearity of Derivatives: The derivative of a sum/difference is the sum/difference of the derivatives.

Step-by-Step Guidance

1. Apply the power rule to each term of the function separately. For each term $a x^n$, the derivative is $a n x^{n-1}$.

2. Compute the derivative of $20x^9$ using the power rule.

3. Compute the derivative of $-4x^7$ using the power rule.

4. Compute the derivative of $8x^5$ using the power rule.

5. Compute the derivative of $-4x$ (remember that $x$ is $x^1$).

6. Combine all the derivative terms to write the full expression for $f'(x)$. Simplify the exponents and coefficients where possible.

Try solving on your own before revealing the answer!

Final Answer:

$f'(x) = 180x^8 - 28x^6 + 40x^4 - 4$

Each term was differentiated using the power rule, and the coefficients and exponents were simplified accordingly.