Question - Differentiation of Mathematical Functions
Solution:
The image is blurry, but it seems to show four mathematical functions that need to be differentiated with respect to $$ x $$. I'll attempt to identify them and differentiate each one:(a) $$ y = (x + n)^1 $$ Using the power rule, we get: $$ \frac{dy}{dx} = 1 \cdot (x + n)^{1 - 1} \cdot \frac{d}{dx}(x + n) $$ $$ \frac{dy}{dx} = (x + n)^0 \cdot \frac{d}{dx}(x + n) $$ Since $$ (x + n)^0 = 1 $$ and $$ \frac{d}{dx}(x + n) = 1 $$, then $$ \frac{dy}{dx} = 1 \cdot 1 = 1 $$(b) $$ f(x) = (2x + n)^1 $$ Applying the power rule: $$ f'(x) = 1 \cdot (2x + n)^{1 - 1} \cdot \frac{d}{dx}(2x + n) $$ $$ f'(x) = (2x + n)^0 \cdot \frac{d}{dx}(2x + n) $$ Since $$ (2x + n)^0 = 1 $$ and $$ \frac{d}{dx}(2x + n) = 2 $$, then $$ f'(x) = 1 \cdot 2 = 2 $$(c) $$ y = (3 - 4x)^5 $$ Using the chain rule, we get: $$ \frac{dy}{dx} = 5(3 - 4x)^{4}(-4) $$ $$ \frac{dy}{dx} = -20(3 - 4x)^{4} $$(d) $$ g(x) = (3z - 4x)^2 $$ (There appears to be a typo in the variable used in the original function. Assuming it's supposed to be $$ x $$, not $$ z $$, and differentiating accordingly:) $$ g'(x) = 2(3 - 4x)^1(-4) $$ $$ g'(x) = -8(3 - 4x) $$If the variables $$ n $$ or $$ z $$ are constants, then my differentiation is correct. If $$ n $$ or $$ z $$ are not constants, and you meant a different variable or power, please provide the correct expressions.
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