Differentiating a function is a fundamental concept in calculus that involves finding the rate of change of a function with respect to its independent variable. The derivative of a function represents the slope of the function at any given point and can be used to analyze various aspects of the function, such as its maximum and minimum values, concavity, and inflection points.
Here are the steps to differentiate a function:
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Identify the function you want to differentiate, typically denoted as $$y = f(x)$$ or $$y = g(t)$$, where $$x$$ or $$t$$ is the independent variable.
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Apply the power rule to differentiate a function. The power rule states that if $$f(x) = x^n$$ or $$x$$ raised to the power $$n$$, then $$f(x) = nx^{(n - 1)}$$ or $$x$$ raised to the power $$(n - 1)$$ and multiplied by $$n$$. For example, if $$f(x) = 5x$$, then $$f'(x) = 5x^{(1 - 1)} = 5$$. Similarly, if $$f(x) = x^{10}$$, then $$f'(x) = 9x^9$$; and if $$f(x) = 2x^5 + x^3 + 10$$, then $$f'(x) = 10x^4 + 3x^2$$.
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Use the sum and difference rules to differentiate a function. The sum and difference rules state that if $$f(x) = u(x) + v(x)$$, then $$f'(x) = u'(x) + v'(x)$$, and if $$f(x) = u(x) - v(x)$$, then $$f'(x) = u'(x) - v'(x)$$. These rules allow you to differentiate functions that are the sum or difference of two or more simpler functions.
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Apply the product rule to differentiate a function. The product rule states that if $$f(x) = u(x) \cdot v(x)$$, then $$f'(x) = u'(x) \cdot v(x) + u(x) \cdot v'(x)$$. This rule is used to differentiate functions that are the product of two simpler functions.
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Use the quotient rule to differentiate a function. The quotient rule states that if $$f(x) = \frac{u(x)}{v(x)}$$, then $$f'(x) = \frac{u'(x) \cdot v(x) - u(x) \cdot v'(x)}{(v(x))^2}$$. This rule is used to differentiate functions that are the quotient of two simpler functions.
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Apply the chain rule to differentiate a composite function. The chain rule allows you to differentiate functions that are composed of two or more simpler functions. If $$y = f(u)$$ and $$u = g(x)$$, then $$\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}$$.
By following these steps, you can find the derivative of a function, which represents its rate of change at any given point. This information is valuable in understanding the behavior of the function and analyzing various aspects of its graph.
