# How do you use definition of derivatives to solve the derivative of #f(x)=–2(sin(x)^5)#?

Solve for the derivative using:

Use the sum of angles trig identity to get

For small angles, we have the following first order approximations

The expected answer.

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To find the derivative of ( f(x) = -2(\sin(x))^5 ), we can use the definition of derivatives. The derivative of a function ( f(x) ) is given by the limit:

[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} ]

Applying this definition to ( f(x) = -2(\sin(x))^5 ), we have:

[ f'(x) = \lim_{h \to 0} \frac{-2(\sin(x + h))^5 + 2(\sin(x))^5}{h} ]

Using the trigonometric identity ( \sin(a + b) = \sin(a)\cos(b) + \cos(a)\sin(b) ), we can expand ( (\sin(x + h))^5 ) as follows:

[ (\sin(x + h))^5 = (\sin(x)\cos(h) + \cos(x)\sin(h))^5 ]

Expanding this using the binomial theorem, we get:

[ (\sin(x + h))^5 = \sin^5(x) + 5\sin^4(x)\cos(x)h + 10\sin^3(x)\cos^2(x)h^2 + 10\sin^2(x)\cos^3(x)h^3 + 5\sin(x)\cos^4(x)h^4 + \cos^5(x)h^5 ]

Substituting this back into the derivative expression and simplifying, we find:

[ f'(x) = \lim_{h \to 0} \frac{-2(\sin(x)\cos(h) + \cos(x)\sin(h))^5 + 2(\sin(x))^5}{h} ]

[ f'(x) = \lim_{h \to 0} \frac{-2(\sin^5(x) + 5\sin^4(x)\cos(x)h + 10\sin^3(x)\cos^2(x)h^2 + 10\sin^2(x)\cos^3(x)h^3 + 5\sin(x)\cos^4(x)h^4 + \cos^5(x)h^5) + 2(\sin(x))^5}{h} ]

[ f'(x) = \lim_{h \to 0} \frac{-10(\sin^4(x)\cos(x)h + \sin^3(x)\cos^2(x)h^2 + 5\sin^2(x)\cos^3(x)h^3 + \sin(x)\cos^4(x)h^4) - 2(\sin(x)\cos(h))^5 + 2(\sin(x))^5}{h} ]

[ f'(x) = -10\sin^4(x)\cos(x) - 2(\sin(x))^5 ]

Therefore, the derivative of ( f(x) = -2(\sin(x))^5 ) is ( f'(x) = -10\sin^4(x)\cos(x) ).

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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