Solve the given problems by finding the appropriate...

Solve the given problems by finding the appropriate derivatives. Find the slope of a line tangent to the curve of the function $y=(4 x+1)\left(x^{4}-1\right)$ at the point $(-1,0) .$ Do not multiply the factors together before taking the derivative. Use the derivative evaluation feature of a calculator to check your result.

Solve the given problems by finding the appropriate derivatives.
Find the slope of a line tangent to the curve of the function $y=(4 x+1)\left(x^{4}-1\right)$ at the point $(-1,0) .$ Do not multiply the factors together before taking the derivative. Use the derivative evaluation feature of a calculator to check your result.

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Key Concepts

Derivative

The derivative of a function represents its instantaneous rate of change and is defined as the limit of the difference quotient as the interval approaches zero. It is the fundamental mathematical tool used for determining the slope of a curve at any given point, thereby providing the slope of the tangent line at that point.

Product Rule

The product rule is a key principle in differentiation used when finding the derivative of the product of two functions. It states that if you have two functions multiplied together, the derivative is found by taking the derivative of the first function and multiplying it by the second, then adding the product of the first function and the derivative of the second.

Tangent Line

A tangent line to a curve at a specific point is a straight line that just 'touches' the curve at that point, sharing the same slope as the curve there. This line provides a linear approximation of the function near that point and plays a crucial role in calculus for understanding the local behavior of functions.

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