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The product rule is a rule for differentiating products of expressions, i.e. 2 expressions multiplied together.
The rule is: if y=A*B, then dy/dx=A*dB/dx +B*dA/dx. This can be remembered as "Write down the first, differentiate the second then write down the second, differentiate the first".
Differentiate y= e2x sin(x) with respect to x.
This is a product rule question because there are 2 expressions multiplied together.
To answer this question, we write down the 2 expressions that have been multiplied together:
A=e2x and B=sin(x)
Then, we differentiate each of them
dA/dx =2e2x (Remember that if you have to differentiate eax, it equals a*eax)
dB/dx=cos(x) (remember that differentiating sin(x) gives cos(x)).
Next, we combine these expressions using the rule at the top
so dy/dx=e2x cos(x) + sin(x) *2e2x
As a final step, we can simplify this by taking out a factor of e2x, so
dy/dx=e2x (cos(x) + 2sin(x)).see more