Use established derivative rules without confusing differentiation with evaluation or multiplication of functions.
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Try
You know from the difference quotient that the derivative of x² is 2x. What should happen to the derivative if every output is multiplied by 3? What if a constant 7 is added to every output? Think about changes in output, not the height of the graph alone.
Predict
A learner claims the derivative of 3x²+7 is 6x+7, keeping the constant. Another claims the derivative of x²·x³ is (2x)(3x²). Predict which parts of these arguments fail before applying a rule.
Experiment
Verify one new power directly. Expand (a+h)³=a³+3a²h+3ah²+h³. Subtract a³ and divide by nonzero h, obtaining 3a²+3ah+h². The limit is 3a². The coefficient and exponent pattern agrees with the derivative of x².
Function
Derivative
Check at x=2
x³
3x²
12
4x³
12x²
48
x³+7
3x²
12
4x³−2x+7
12x²−2
46
Sketch x³ and x³+7 using the same scale. Their vertical positions differ, but corresponding secant slopes agree because the added constants cancel in each numerator.
Understand
The power rule gives the derivative of x to power n as n times x to power n−1 wherever the applicable real power function is differentiable. For positive integer n it holds for all real x. For negative powers, exclude zero. For general real powers, positive x is a safe domain; any endpoint or negative-input extension needs its own domain and differentiability check.
The derivative of a constant is zero. The derivative of c f(x), where c is fixed, is c f′(x). Derivatives of sums and differences are the corresponding sums and differences of derivatives, provided the terms are differentiable. These rules follow from the limit laws applied to difference quotients.
For 5/x², first write 5x^(−2). Its derivative is −10x^(−3), or −10/x³, with x≠0. For √x, write x^(1/2); the derivative is 1/(2√x) for positive x. At zero, the right difference quotient is 1/√h and is unbounded. The displayed formula must not be extended by pretending a denominator zero is allowed.
The product shortcut is false. Combining x²·x³ first gives x⁵, whose derivative is 5x⁴, not 6x³. A later lesson will teach the general product rule. Rules for linear combinations do not automatically extend to products, quotients or compositions.
For f(x)=x³−12x, setting f′(x)=3x²−12 equal to zero gives candidates x=−2 and x=2. These are stationary inputs. The equation alone does not classify a maximum or minimum or establish a global optimum; those conclusions need further analysis.
Apply
Differentiate symbolically before substituting a requested input. Keep function values and derivative values on separate lines. In the finite-set task, show why the factorization gives every solution. For the fractional-power task, explicitly state the domain and discuss the endpoint rather than reporting only a formula.
Continue
Use these derivatives to construct local predictions in the final lesson of this tranche. Product, quotient, general chain, trigonometric and exponential differentiation remain further course work; this lesson does not claim full differentiation fluency.
Apply · explain · revise
Practice and reasoning
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For f(x)=3x⁴−2x²+7, find f′(−1).
Integers, fractions and finite decimals are accepted. Expressions such as sqrt(2) are not parsed by this checker.
For f(x)=5/x² with x≠0, find f′(2).
Integers, fractions and finite decimals are accepted. Expressions such as sqrt(2) are not parsed by this checker.
Find all real inputs where the derivative of f(x)=x³−12x is zero. Enter a finite set.
Integers, fractions and finite decimals are accepted. Separate values with commas; type none for no solutions. Expressions such as sqrt(2) are not parsed by this checker.
Differentiate f(x)=√x for x>0, then explain why that formula does not provide a finite derivative at x=0.
Structure your argument and preserve revisions
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Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
A learner differentiates x²·x³ by multiplying derivatives and gets 6x³. Which repair is valid using only power rules?
Reflect and continue
Before marking this lesson complete, explain one key idea without the worked solution and identify a condition that matters. Marking complete records participation, not independent mastery.
