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Derivatives, antiderivatives and indefinite integrals
A function \( F(x) \) has a derivative \( F'(x) \).
What exactly is the function \( F'(x) \)?
Is \(F'(x)=\sin(x)\)?, or \(3 \sqrt{x}\)?, or,... what? \)?
\( F'(x) = \)? \( F'(x) = f(x) \)
Example 1.Example 1. Example 1.
Let \( F(x)=3x^5 + 14 \). Find the derivative \( F'(x) \).
Let \( F(x)=3x^5 + 14 \). Find the derivative \( F'(x) \).
By the power rule, \( F'(x)=(3\cdot 5) x^4 + 0 \),
By the power rule, \( F'(x)=(3\cdot 5) x^4 + 0 \), and so
which reduces to \( F'(x)= 15 x^4 \).
Example 2.Example 2.
Let \( f(x)=15 x^4 \).
What function has \( f(x) \) as its derivative?
Let \( f(x)=15 x^4 \).
Let \( f(x)=15 x^4\).
What function has \( f(x) \) as its derivative?
The derivative of \( 3 x^5 + C \) is \(3\cdot 5 \cdot x^{5-1} + 0 = f(x) \).
The derivative of \( 3 x^5 + C \) is \( 3\cdot 5 \cdot x^{5-1} + 0 = f(x)\).
That makes \( 3 x^5 + C \) the antiderivative of \( f(x) \).
That is, \(F(x)= 3 x^5 + C \).
Example 3.Example 3.
What is the integral of \( 15 x^4 \)?
What is the integral of \( 15 x^4 \)?
What is \( \displaystyle \int 15 x^4\, dx = ? \) What is \( \displaystyle \int 15 x^4\, dx = ? \)
This thing,
\(\rightarrow \displaystyle \int 15 x^4\, dx \),
is called the indefinite integral of \( 15 x^4 \),
and is also known as the antiderivative of \( 15 x^4 \).
So, (by Example 2) the integral \[ \displaystyle \int 15 x^4\, dx = 3 x^5 + C . \]