
Find the derivative of , by using the first principle of derivatives.
Answer
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Hint: We recall the first principle of derivative. We assume a small change in as and its corresponding change in as . We find the average rate of change as . We take limit to find the instantaneous rate of change as derivative of .
Complete step-by-step solution:
We are given the function in the question. Let us have . Let be a very small change in and the corresponding change in be . So we have;
We subtract both sides of the above equation to have;
We divide both sides of the above step to have;
We take limit both sides of the above step to have;
We use the trigonometric identity for in the above step to have;
We use law of product if limits in the right hand side of the above step to have;
We use the standard limit for in the right hand side of the above step to have;
We use the double angle formula for in the right hand side of the above step to have;
We know from first principle of derivative that . So we have
Note: We can use chain rule to directly find the derivative of . If composite function is defined as then the chain rule is given as . We can also use the first principle for derivative with a very small change as . The derivative of the function at particular points geometrically gives the slope of the tangent to the curve of the function. The first principle is also known as the delta method.
Complete step-by-step solution:
We are given the function
We subtract
We divide
We take limit
We use the trigonometric identity
We use law of product if limits in the right hand side of the above step to have;
We use the standard limit
We use the double angle formula
We know from first principle of derivative that
Note: We can use chain rule to directly find the derivative of
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