Calculus
⏱ ~3-min readAceMark Guide
What this topic is really about
Applying the chain rule, we first differentiate the outer sine function to get cos(x^2) and then multiply by the derivative of the inner function x^2, which is 2x. Option A is incorrect because it fails to apply the chain rule, neglecting the derivative of the inner function entirely.
Using the power rule for integration, we increase the exponent of x by one and divide by the new exponent, resulting in x^2/2 plus an integration constant C. Option B is incorrect because it fails to divide by the new exponent and omits the arbitrary constant C.
See the mechanism
Applying the chain rule, we first differentiate the outer sine function to get cos(x^2) and then multiply by the derivative of the inner function x^2, which is 2x. A diagram for this topic isn't available yet — the worked example below walks the same reasoning step by step.
An exam-style question, fully explained
The derivative of sin(x²) with respect to x is:
- Identify what the question tests: The derivative of sin(x²) with respect to x is:.
- Applying the chain rule, we first differentiate the outer sine function to get cos(x^2) and then multiply by the derivative of the inner function x^2, which is 2x.
- Option A is incorrect because it fails to apply the chain rule, neglecting the derivative of the inner function entirely.
Traps the examiner sets
- Option A is incorrect because it fails to apply the chain rule, neglecting the derivative of the inner function entirely.
- Option B is incorrect because it fails to divide by the new exponent and omits the arbitrary constant C.
- Option B is incorrect because it incorrectly applies the power rule, which is only valid for variable bases with constant exponents.
Test your recall
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