The line tangent to the curve y=h(x) at x=4 is y = −3x+14. Find an equation of the line tangent to the following curves at x=4.
y = (x²-3x)h(x)

Verified step by step guidance

1. Step 1: Identify the given information. We know that the line tangent to the curve y = h(x) at x = 4 is y = -3x + 14. This implies that h(4) is the y-coordinate of the point of tangency, and the derivative h'(4) is the slope of the tangent line, which is -3.
2. Step 2: Use the product rule to find the derivative of y = (x^2 - 3x)h(x). The product rule states that if y = u(x)v(x), then y' = u'(x)v(x) + u(x)v'(x). Here, u(x) = x^2 - 3x and v(x) = h(x).
3. Step 3: Differentiate u(x) = x^2 - 3x. The derivative u'(x) is 2x - 3.
4. Step 4: Apply the product rule. Substitute u(x), u'(x), v(x), and v'(x) into the product rule formula: y' = (2x - 3)h(x) + (x^2 - 3x)h'(x).
5. Step 5: Evaluate the derivative at x = 4. Substitute x = 4, h(4), and h'(4) = -3 into the derivative expression to find the slope of the tangent line at x = 4. Use this slope and the point (4, (4^2 - 3*4)h(4)) to write the equation of the tangent line in point-slope form: y - y_1 = m(x - x_1).

Key Concepts

Tangent Line

A tangent line to a curve at a given point is a straight line that touches the curve at that point without crossing it. The slope of the tangent line represents the instantaneous rate of change of the function at that point, which is given by the derivative. In this case, the equation of the tangent line is provided, allowing us to determine the slope and use it for further calculations.

Product Rule

The Product Rule is a fundamental differentiation rule used when finding the derivative of the product of two functions. It states that if you have two functions u(x) and v(x), the derivative of their product is given by u'v + uv'. This rule is essential for differentiating the function y = (x² - 3x)h(x) since it involves the product of the polynomial and the function h(x).

Chain Rule

The Chain Rule is a technique for differentiating composite functions. It states that if a function y is composed of another function u, then the derivative of y with respect to x can be found by multiplying the derivative of y with respect to u by the derivative of u with respect to x. This rule may be relevant if h(x) itself is a composite function, requiring careful application when differentiating at x=4.