Pure: Calculus

Applying the power rule, we multiply by the exponent and decrease the power by one, which gives 12x^3 for the first term and -10x for the second. The constant 7 differentiates to zero, making option A correct. Option B is incorrect because it fails to multiply the coefficient of the second term by its original exponent of 2.

According to the chain rule, we differentiate the outer sine function to get cosine, and then multiply by the derivative of the inner function 2x, which is 2. This yields 2cos(2x), making option B correct. Option A is incorrect because it fails to apply the chain rule and leaves out the factor of 2.

Applying the power rule, we multiply by the exponent and decrease the power by one, which gives 12x^3 for the first term and -10x for the second. A diagram for this topic isn't available yet — the worked example below walks the same reasoning step by step.

• !Option B is incorrect because it fails to multiply the coefficient of the second term by its original exponent of 2.
• !Option D is incorrect because it shows differentiation rather than integration, finding the derivative instead of the antiderivative.
• !Option A is incorrect because it fails to apply the chain rule and leaves out the factor of 2.
• !Option D is incorrect because e^x is the derivative of the exponential function e^x, rather than the logarithmic function.

Answer each from memory — you'll see instantly whether you're right and why.

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