Find the derivative if f(x)= e^-3x times sin(4x) use the product rule.
d/dx [e^-(3x) ] [ sin (4 x) ]
= [e^-(3x) ] d/dx [ sin (4 x) ]
+ [ sin (4 x) ] d/dx [e^-(3x) ]
= [e^-(3x) ][4 cos (4x) ]
+ [ sin (4 x) ][ -3 e^-(3x) ]
= e^-(3x)[ 4 cos (4x) -3 sin (4x) ]
To find the derivative of the function f(x) = e^(-3x) * sin(4x) using the product rule, you need to apply the following steps:
Step 1: Identify the two functions being multiplied.
In this case, the two functions being multiplied are e^(-3x) and sin(4x).
Step 2: Differentiate the first function.
Differentiating e^(-3x) with respect to x gives us -3e^(-3x).
Step 3: Keep the second function as it is.
Leave sin(4x) unchanged.
Step 4: Differentiate the second function.
Differentiating sin(4x) with respect to x gives us 4cos(4x).
Step 5: Apply the product rule formula.
The product rule states that the derivative of the product of two functions u(x) and v(x) is given by:
d(uv)/dx = u * dv/dx + v * du/dx
Applying the product rule, we have:
f'(x) = (e^(-3x) * 4cos(4x)) + (sin(4x) * -3e^(-3x))
Step 6: Simplify the expression.
f'(x) = 4e^(-3x)cos(4x) - 3e^(-3x)sin(4x)
So, the derivative of f(x) = e^(-3x) * sin(4x) using the product rule is f'(x) = 4e^(-3x)cos(4x) - 3e^(-3x)sin(4x).