Differentiate sin2x w.r.t. ecosx.
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Step-by-Step Solution
Step 1: Define functions
To differentiate one function with respect to another, we first define each function separately. Let u=sin2x and v=ecosx. We need to find dvdu.
Step 2: Differentiate u with respect to x
Now, we differentiate u with respect to x. Using the chain rule, the derivative of sin2x is 2sinx⋅dxd(sinx)=2sinxcosx.
Step 3: Differentiate v with respect to x
Next, we differentiate v with respect to x. Using the chain rule, the derivative of ecosx is ecosx⋅dxd(cosx)=ecosx(−sinx)=−ecosxsinx.
Step 4: Find du/dv
To find dvdu, we use the formula dvdu=dv/dxdu/dx. We substitute the expressions we found for dxdu and dxdv.
Step 5: Simplify the expression
We can simplify the expression by canceling out sinx from the numerator and the denominator, assuming sinx=0. This gives us the final derivative.