Math Problem Statement
Find the derivative of: 7√x − 3/x³ − 3x + 6
Solution
The given expression is:
We need to find the derivative .
Step-by-step solution:
- Rewrite the terms in exponent form:
- becomes
- becomes
- The term remains as is.
- The constant remains unchanged (its derivative will be 0).
So the function now looks like:
-
Differentiate each term:
- For , apply the power rule:
- For , apply the power rule again:
- For , the derivative is simply:
- The derivative of the constant is 0.
-
Combine the results:
Thus, the derivative of the given function is:
Would you like more details on any specific part of this solution?
5 Related Questions:
- How do you apply the power rule to expressions with fractional exponents?
- What is the derivative of a cube root function like ?
- How can you differentiate rational expressions like ?
- Why is the derivative of a constant always zero?
- Can you explain the general rules for differentiating terms with negative exponents?
Tip:
When differentiating functions with roots or fractional exponents, always rewrite them in exponent form to simplify the application of the power rule.
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Math Problem Analysis
Mathematical Concepts
Differentiation
Power Rule
Fractional Exponents
Negative Exponents
Formulas
Power rule: d/dx(x^n) = n*x^(n-1)
Fractional exponent rule: d/dx(x^(m/n)) = (m/n)*x^((m/n)-1)
Negative exponent rule: d/dx(x^(-n)) = -n*x^(-n-1)
Theorems
Power Rule
Suitable Grade Level
Grades 11-12 (Calculus)