Learn how to find the derivative of x^3 + 5x^2. The derivative is d/dx[x^3 + 5x^2].
Below is the graph of f(x) = x^3 + 5x^2 and its derivative f'(x). Notice how the original function and derivative relate to each other.
Original Function:
Derivative:
This derivative represents the rate of change of the original function. The calculation follows standard differentiation rules as shown in the step-by-step solution below.
The expression x^3 + 5x^2 is a sum of two terms, each of which is a polynomial term. The Power Rule applies to terms that involve powers of x, so we will use this rule for both x^3 and 5x^2.
💡 Why this works: Each term in this expression is of the form ax^n, where 'a' is a constant and 'n' is the exponent. The Power Rule is ideal for differentiating such terms, allowing us to differentiate each term separately.
We apply the Power Rule to each term separately. For the term x^3, the Power Rule tells us to multiply by 3 and reduce the exponent by 1, yielding 3x^2. For the term 5x^2, we multiply by 2 and reduce the exponent by 1, giving 10x.
💡 Why this works: The differentiation of x^3 results in 3x^2, and the differentiation of 5x^2 gives 10x. Combining these results gives the final derivative, f'(x) = 3x^2 + 10x.
The simplified form of the derivative is f'(x) = 3x^2 + 10x. This is the final result after applying the Power Rule to each term.
💡 Why this works: By applying the Power Rule to both terms and simplifying, we arrive at the final expression for the derivative. This form is correct, and we can verify it by checking that the derivative describes the slope of the original function at any point.
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❌ Common Mistake:
Not simplifying the final answer
✅ Correct Approach:
Always simplify your derivative to its most reduced form
Double-check your work by comparing your steps with our solution. Use our calculator to verify each step of the differentiation process.
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