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If y=(u+4)/(u-4 ) and u=(x+√x)/10 find dy/dx when x=4.
Mar 15, 2024
If y=(u+4)/(u-4 ) and u=(x+√x)/10 find dy/dx when x=4.
Generated Graph
Solution by Steps
step 1
To find dydx \frac{dy}{dx} when x=4 x=4 , we first need to differentiate y y with respect to u u using the given result from asksia-ll calculator
step 2
The derivative of y y with respect to u u is dydu=−8(u−4)2 \frac{dy}{du} = -\frac{8}{(u - 4)^2}
step 3
Next, we differentiate u u with respect to x x using the given result from asksia-ll calculator
step 4
The derivative of u u with respect to x x is dudx=120(1x+2) \frac{du}{dx} = \frac{1}{20} \left(\frac{1}{\sqrt{x}} + 2\right)
step 5
Now, we apply the chain rule to find dydx \frac{dy}{dx} by multiplying dydu \frac{dy}{du} and dudx \frac{du}{dx}
step 6
Substituting the derivatives from steps 2 and 4, we get dydx=dydu⋅dudx=−8(u−4)2⋅120(1x+2) \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} = -\frac{8}{(u - 4)^2} \cdot \frac{1}{20} \left(\frac{1}{\sqrt{x}} + 2\right)
step 7
Substitute u=x+x10 u = \frac{x + \sqrt{x}}{10} into dydx \frac{dy}{dx}
step 8
Finally, substitute x=4 x = 4 into the expression for dydx \frac{dy}{dx} to find the value when x=4 x = 4
step 9
After substitution, we simplify to get the final value of dydx \frac{dy}{dx} when x=4 x = 4
Answer
[Insert final answer here]
Key Concept
Chain Rule in Differentiation
Explanation
To find dydx \frac{dy}{dx} , we use the chain rule to combine the derivatives dydu \frac{dy}{du} and dudx \frac{du}{dx} , and then evaluate at the given value of x x .
1. Differentiate each function. [3 marks each, K/UK / U ] a) g(x)=(4x2+2x−3)5g(x)=\left(4 x^{2}+2 x-3\right)^{5} b) y=x2−2x3y=\sqrt[3]{x^{2}-2 x} c) g(x)=(2x2+7x−6)−4g(x)=\left(2 x^{2}+7 x-6\right)^{-4}
Generated Graph
Solution by Steps
step 1
Apply the chain rule to differentiate (4x2+2x−3)5 (4x^{2}+2x-3)^{5}
step 2
The derivative of the outer function raised to the 5th power is 5(4x2+2x−3)4 5(4x^{2}+2x-3)^{4}
step 3
Multiply by the derivative of the inner function, 8x+2 8x+2
step 4
The final derivative is 5(8x+2)(4x2+2x−3)4 5(8x+2)(4x^{2}+2x-3)^{4}
Answer
ddxg(x)=5(8x+2)(4x2+2x−3)4 \frac{d}{dx}g(x) = 5(8x+2)(4x^{2}+2x-3)^{4}
Key Concept
Chain Rule
Explanation
The chain rule is used when differentiating a composite function, which in this case is a function raised to a power.
For question b) y=x2−2x3 y=\sqrt[3]{x^{2}-2x}
step 1
Apply the chain rule to differentiate (x2−2x)13 (x^{2}-2x)^{\frac{1}{3}}
step 2
The derivative of the outer function with respect to the inner function is 13(x2−2x)−23 \frac{1}{3}(x^{2}-2x)^{-\frac{2}{3}}
step 3
Multiply by the derivative of the inner function, 2x−2 2x-2
step 4
Simplify to get the final derivative 2(x−1)3(x2−2x)23 \frac{2(x-1)}{3(x^{2}-2x)^{\frac{2}{3}}}
Answer
ddxy=2(x−1)3(x2−2x)23 \frac{d}{dx}y = \frac{2(x-1)}{3(x^{2}-2x)^{\frac{2}{3}}}
Key Concept
Chain Rule
Explanation
The chain rule is applied to the power of a binomial, which is then multiplied by the derivative of the binomial itself.
For question c) g(x)=(2x2+7x−6)−4 g(x)=(2x^{2}+7x-6)^{-4}
step 1
Apply the chain rule to differentiate (2x2+7x−6)−4 (2x^{2}+7x-6)^{-4}
step 2
The derivative of the outer function raised to the −4-4th power is −4(2x2+7x−6)−5 -4(2x^{2}+7x-6)^{-5}
step 3
Multiply by the derivative of the inner function, 4x+7 4x+7
step 4
The final derivative is −4(4x+7)(2x2+7x−6)5 -\frac{4(4x+7)}{(2x^{2}+7x-6)^{5}}
Answer
ddxg(x)=−4(4x+7)(2x2+7x−6)5 \frac{d}{dx}g(x) = -\frac{4(4x+7)}{(2x^{2}+7x-6)^{5}}
Key Concept
Chain Rule
Explanation
The chain rule is used to differentiate a function raised to a negative power, which involves multiplying the negative power by the derivative of the function inside the power.
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