Herrera Tyler
07/17/2023 · Middle School
b) Using the \( 1^{\text {st }} \) principal differentiate \( y=4 x^{2}+5 x-3 \) and determine the gradient of the curve at \( x=-3 \)
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Step-by-step Solution
Find the first order derivative with respect to \( x \) for \( 4x^2 + 5x - 3 \).
Evaluate the derivative by following steps:
- step0: Evaluate the derivative:
\(\frac{d}{dx}\left(4x^{2}+5x-3\right)\)
- step1: Use differentiation rules:
\(\frac{d}{dx}\left(4x^{2}\right)+\frac{d}{dx}\left(5x\right)-\frac{d}{dx}\left(3\right)\)
- step2: Calculate:
\(8x+5-0\)
- step3: Remove 0:
\(8x+5\)
Substitute \( x=-3 \) into the expression \( 8x+5 \).
Evaluate the expression by following steps:
- step0: Evaluate:
\(8x+5\)
- step1: Substitute:
\(8\left(-3\right)+5\)
- step2: Multiply the numbers:
\(-24+5\)
- step3: Add the numbers:
\(-19\)
The first principal derivative of the function \( y=4x^{2}+5x-3 \) is \( 8x+5 \).
Substituting \( x=-3 \) into the derivative, we find that the gradient of the curve at \( x=-3 \) is -19.
Quick Answer
The gradient of the curve at \( x=-3 \) is -19.
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