Find the equation of the line tangent to the given curve at...

Find the equation of the line tangent to the given curve at the specified value of $x .$ Express the equation exactly in the form $y=m x+c$
a) $y=\left(2 x^{2}-1\right)^{3} \quad x=-1$
b) $y=\sqrt{3 x^{2}-2} \quad x=3$
c) $y=\sin 2 x$ $x=\pi$
d) $y=\frac{x^{3}+1}{2 x} \quad x=1$

Key Concepts

Derivative

The derivative of a function represents its instantaneous rate of change and provides the slope of the tangent line at any given point. It is computed using various differentiation techniques.

Tangent Line

The tangent line to a curve at a given point is the line that best approximates the curve near that point. It has a slope equal to the derivative of the function at that point, and it touches the curve without crossing it locally.

Differentiation Rules

Various rules, such as the chain rule, product rule, and quotient rule, are used to differentiate complex functions. These rules simplify the process of finding the derivative of functions composed of multiple terms or nested functions.

Chain Rule

The chain rule is a method for finding the derivative of composite functions. When a function is composed of two or more functions, the derivative of the outer function is found and then multiplied by the derivative of the inner function.

Quotient Rule

The quotient rule is used to differentiate functions that are expressed as one function divided by another. It provides a systematic way to compute the derivative of a ratio of two differentiable functions.

Slope-Intercept Form

The slope-intercept form of a linear equation, written as y = mx + c, expresses the line in terms of its slope m and its y-intercept c. This form is often used to present the final equation of a tangent line.

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