y=a^(2)Arcsin((x)/(a))-xsqrt(a^(2)-x^(2))quad(a= constant )

Find Derivative of Function: We need to find the derivative of the function y with respect to x. The function is y = a^2 * arcsin(x/a) - x * sqrt(a^2 - x^2), where a is a constant. We will use the chain rule, product rule, and the derivatives of basic functions to find the derivative.Derivative of First Term: First, let's find the derivative of the first term a^2 * arcsin(x/a). Since a is a constant, we can treat a^2 as a constant multiplier. The derivative of arcsin(u) with respect to u is 1/sqrt(1-u^2). Here, u = x/a, so we need to apply the chain rule.Derivative of Second Term: The derivative of arcsin(x/a) with respect to x is (1/sqrt(1-(x/a)^2)) * (d/dx)(x/a). Since (d/dx)(x/a) = 1/a, the derivative of the first term is a^2 * (1/sqrt(1-(x/a)^2)) * (1/a).Combine Derivatives: Simplifying the derivative of the first term, we get (a^2/a) * (1/sqrt(1-(x/a)^2)) = a/sqrt(a^2 - x^2).Simplify Final Derivative: Now, let's find the derivative of the second term -x * sqrt(a^2 - x^2). We will use the product rule, which states that (d/dx)(u*v) = u' * v + u * v', where u = -x and v = sqrt(a^2 - x^2).The derivative of -x with respect to x is -1. Now we need to find the derivative of sqrt(a^2 - x^2) with respect to x. We will use the chain rule again, where the outer function is the square root and the inner function is (a^2 - x^2).The derivative of sqrt(u) with respect to u is (1/2sqrt(u)). The derivative of (a^2 - x^2) with respect to x is -2x. Applying the chain rule, the derivative of sqrt(a^2 - x^2) is (1/2sqrt(a^2 - x^2)) * (-2x).Simplifying the derivative of sqrt(a^2 - x^2), we get (-2x)/(2sqrt(a^2 - x^2)) = -x/sqrt(a^2 - x^2).Applying the product rule to the second term -x * sqrt(a^2 - x^2), we get -1 * sqrt(a^2 - x^2) + (-x) * (-x/sqrt(a^2 - x^2)).Simplifying the result from the product rule, we get -sqrt(a^2 - x^2) + x^2/sqrt(a^2 - x^2).Now we combine the derivatives of the two terms to find the derivative of the entire function y. The derivative of y with respect to x is a/sqrt(a^2 - x^2) - sqrt(a^2 - x^2) + x^2/sqrt(a^2 - x^2).We can simplify the expression by combining the terms over the common denominator sqrt(a^2 - x^2). The final derivative of y with respect to x is (a - (a^2 - x^2) + x^2) / sqrt(a^2 - x^2).After simplifying the numerator, we notice that the x^2 terms cancel out, and we are left with a / sqrt(a^2 - x^2). This is the final derivative of the function y with respect to x.

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y=a^(2)Arcsin((x)/(a))-xsqrt(a^(2)-x^(2))quad(a= constant
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$y=a^{2}\arcsin\left(\frac{x}{a}\right)-x\sqrt{a^{2}-x^{2}}\quad(a=\text{constant})$

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Find derivatives of using multiple formulae

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y=a^{2}\arcsin\left(\frac{x}{a}\right)-x\sqrt{a^{2}-x^{2}}\quad(a=\text{constant})

Find Derivative of Function: We need to find the derivative of the function $y$ with respect to $x$ . The function is $y = a^2 \cdot \arcsin(\frac{x}{a}) - x \cdot \sqrt{a^2 - x^2}$ , where $a$ is a constant. We will use the chain rule, product rule, and the derivatives of basic functions to find the derivative.
Derivative of First Term: First, let's find the derivative of the first term $a^2 \cdot \arcsin(\frac{x}{a})$ . Since $a$ is a constant, we can treat $a^2$ as a constant multiplier. The derivative of $\arcsin(u)$ with respect to $u$ is $\frac{1}{\sqrt{1-u^2}}$ . Here, $u = \frac{x}{a}$ , so we need to apply the chain rule.
Derivative of Second Term: The derivative of $\arcsin(\frac{x}{a})$ with respect to $x$ is $\frac{1}{\sqrt{1-(\frac{x}{a})^2}}$ * $\frac{d}{dx}(\frac{x}{a})$ . Since $\frac{d}{dx}(\frac{x}{a}) = \frac{1}{a}$ , the derivative of the first term is $a^2 * \frac{1}{\sqrt{1-(\frac{x}{a})^2}} * \frac{1}{a}$ .
Combine Derivatives: Simplifying the derivative of the first term, we get $(a^2/a) \times (1/\sqrt{1-(x/a)^2}) = a/\sqrt{a^2 - x^2}$ .
Simplify Final Derivative: Now, let's find the derivative of the second term $-x \cdot \sqrt{a^2 - x^2}$ . We will use the product rule, which states that $\frac{d}{dx}(u \cdot v) = u' \cdot v + u \cdot v'$ , where $u = -x$ and $v = \sqrt{a^2 - x^2}$ .
Simplify Final Derivative: Now, let's find the derivative of the second term $-x \sqrt{a^2 - x^2}$ . We will use the product rule, which states that $(\frac{d}{dx})(u\cdot v) = u' \cdot v + u \cdot v'$ , where $u = -x$ and $v = \sqrt{a^2 - x^2}$ .The derivative of $-x$ with respect to $x$ is $-1$ . Now we need to find the derivative of $\sqrt{a^2 - x^2}$ with respect to $x$ . We will use the chain rule again, where the outer function is the square root and the inner function is $(a^2 - x^2)$ .
Simplify Final Derivative: Now, let's find the derivative of the second term $-x \sqrt{a^2 - x^2}$ . We will use the product rule, which states that $(\frac{d}{dx})(u*v) = u' * v + u * v'$ , where $u = -x$ and $v = \sqrt{a^2 - x^2}$ .The derivative of $-x$ with respect to $x$ is $-1$ . Now we need to find the derivative of $\sqrt{a^2 - x^2}$ with respect to $x$ . We will use the chain rule again, where the outer function is the square root and the inner function is $(a^2 - x^2)$ .The derivative of $(\frac{d}{dx})(u*v) = u' * v + u * v'$ $0$ with respect to $(\frac{d}{dx})(u*v) = u' * v + u * v'$ $1$ is $(\frac{d}{dx})(u*v) = u' * v + u * v'$ $2$ . The derivative of $(a^2 - x^2)$ with respect to $x$ is $(\frac{d}{dx})(u*v) = u' * v + u * v'$ $5$ . Applying the chain rule, the derivative of $\sqrt{a^2 - x^2}$ is $(\frac{d}{dx})(u*v) = u' * v + u * v'$ $7$ .
Simplify Final Derivative: Now, let's find the derivative of the second term $-x \cdot \sqrt{a^2 - x^2}$ . We will use the product rule, which states that $\frac{d}{dx}(u\cdot v) = u' \cdot v + u \cdot v'$ , where $u = -x$ and $v = \sqrt{a^2 - x^2}$ .The derivative of $-x$ with respect to $x$ is $-1$ . Now we need to find the derivative of $\sqrt{a^2 - x^2}$ with respect to $x$ . We will use the chain rule again, where the outer function is the square root and the inner function is $(a^2 - x^2)$ .The derivative of $\frac{d}{dx}(u\cdot v) = u' \cdot v + u \cdot v'$ $0$ with respect to $\frac{d}{dx}(u\cdot v) = u' \cdot v + u \cdot v'$ $1$ is $\frac{d}{dx}(u\cdot v) = u' \cdot v + u \cdot v'$ $2$ . The derivative of $(a^2 - x^2)$ with respect to $x$ is $\frac{d}{dx}(u\cdot v) = u' \cdot v + u \cdot v'$ $5$ . Applying the chain rule, the derivative of $\sqrt{a^2 - x^2}$ is $\frac{d}{dx}(u\cdot v) = u' \cdot v + u \cdot v'$ $7$ .Simplifying the derivative of $\sqrt{a^2 - x^2}$ , we get $\frac{d}{dx}(u\cdot v) = u' \cdot v + u \cdot v'$ $9$ .
Simplify Final Derivative: Now, let's find the derivative of the second term $-x \sqrt{a^2 - x^2}$ . We will use the product rule, which states that $(d/dx)(u*v) = u' * v + u * v'$ , where $u = -x$ and $v = \sqrt{a^2 - x^2}$ .The derivative of $-x$ with respect to $x$ is $-1$ . Now we need to find the derivative of $\sqrt{a^2 - x^2}$ with respect to $x$ . We will use the chain rule again, where the outer function is the square root and the inner function is $(a^2 - x^2)$ .The derivative of $(d/dx)(u*v) = u' * v + u * v'$ $0$ with respect to $(d/dx)(u*v) = u' * v + u * v'$ $1$ is $(d/dx)(u*v) = u' * v + u * v'$ $2$ . The derivative of $(a^2 - x^2)$ with respect to $x$ is $(d/dx)(u*v) = u' * v + u * v'$ $5$ . Applying the chain rule, the derivative of $\sqrt{a^2 - x^2}$ is $(d/dx)(u*v) = u' * v + u * v'$ $7$ .Simplifying the derivative of $\sqrt{a^2 - x^2}$ , we get $(d/dx)(u*v) = u' * v + u * v'$ $9$ .Applying the product rule to the second term $u = -x$ $0$ , we get $u = -x$ $1$ .
Simplify Final Derivative: Now, let's find the derivative of the second term $-x \sqrt{a^2 - x^2}$ . We will use the product rule, which states that $(d/dx)(u*v) = u' * v + u * v'$ , where $u = -x$ and $v = \sqrt{a^2 - x^2}$ .The derivative of $-x$ with respect to $x$ is $-1$ . Now we need to find the derivative of $\sqrt{a^2 - x^2}$ with respect to $x$ . We will use the chain rule again, where the outer function is the square root and the inner function is $(a^2 - x^2)$ .The derivative of $(d/dx)(u*v) = u' * v + u * v'$ $0$ with respect to $(d/dx)(u*v) = u' * v + u * v'$ $1$ is $(d/dx)(u*v) = u' * v + u * v'$ $2$ . The derivative of $(a^2 - x^2)$ with respect to $x$ is $(d/dx)(u*v) = u' * v + u * v'$ $5$ . Applying the chain rule, the derivative of $\sqrt{a^2 - x^2}$ is $(d/dx)(u*v) = u' * v + u * v'$ $7$ .Simplifying the derivative of $\sqrt{a^2 - x^2}$ , we get $(d/dx)(u*v) = u' * v + u * v'$ $9$ .Applying the product rule to the second term $u = -x$ $0$ , we get $u = -x$ $1$ + $u = -x$ $2$ .Simplifying the result from the product rule, we get $u = -x$ $3$ .
Simplify Final Derivative: Now, let's find the derivative of the second term $-x \sqrt{a^2 - x^2}$ . We will use the product rule, which states that $(d/dx)(u\cdot v) = u' \cdot v + u \cdot v'$ , where $u = -x$ and $v = \sqrt{a^2 - x^2}$ .The derivative of $-x$ with respect to $x$ is $-1$ . Now we need to find the derivative of $\sqrt{a^2 - x^2}$ with respect to $x$ . We will use the chain rule again, where the outer function is the square root and the inner function is $(a^2 - x^2)$ .The derivative of $(d/dx)(u\cdot v) = u' \cdot v + u \cdot v'$ $0$ with respect to $(d/dx)(u\cdot v) = u' \cdot v + u \cdot v'$ $1$ is $(d/dx)(u\cdot v) = u' \cdot v + u \cdot v'$ $2$ . The derivative of $(a^2 - x^2)$ with respect to $x$ is $(d/dx)(u\cdot v) = u' \cdot v + u \cdot v'$ $5$ . Applying the chain rule, the derivative of $\sqrt{a^2 - x^2}$ is $(d/dx)(u\cdot v) = u' \cdot v + u \cdot v'$ $7$ .Simplifying the derivative of $\sqrt{a^2 - x^2}$ , we get $(d/dx)(u\cdot v) = u' \cdot v + u \cdot v'$ $9$ .Applying the product rule to the second term $u = -x$ $0$ , we get $u = -x$ $1$ + $u = -x$ $2$ .Simplifying the result from the product rule, we get $u = -x$ $3$ .Now we combine the derivatives of the two terms to find the derivative of the entire function $u = -x$ $4$ . The derivative of $u = -x$ $4$ with respect to $x$ is $u = -x$ $7$ .
Simplify Final Derivative: Now, let's find the derivative of the second term $-x \cdot \sqrt{a^2 - x^2}$ . We will use the product rule, which states that $\frac{d}{dx}(u\cdot v) = u' \cdot v + u \cdot v'$ , where $u = -x$ and $v = \sqrt{a^2 - x^2}$ .The derivative of $-x$ with respect to $x$ is $-1$ . Now we need to find the derivative of $\sqrt{a^2 - x^2}$ with respect to $x$ . We will use the chain rule again, where the outer function is the square root and the inner function is $(a^2 - x^2)$ .The derivative of $\frac{d}{dx}(u\cdot v) = u' \cdot v + u \cdot v'$ $0$ with respect to $\frac{d}{dx}(u\cdot v) = u' \cdot v + u \cdot v'$ $1$ is $\frac{d}{dx}(u\cdot v) = u' \cdot v + u \cdot v'$ $2$ . The derivative of $(a^2 - x^2)$ with respect to $x$ is $\frac{d}{dx}(u\cdot v) = u' \cdot v + u \cdot v'$ $5$ . Applying the chain rule, the derivative of $\sqrt{a^2 - x^2}$ is $\frac{d}{dx}(u\cdot v) = u' \cdot v + u \cdot v'$ $7$ .Simplifying the derivative of $\sqrt{a^2 - x^2}$ , we get $\frac{d}{dx}(u\cdot v) = u' \cdot v + u \cdot v'$ $9$ .Applying the product rule to the second term $-x \cdot \sqrt{a^2 - x^2}$ , we get $u = -x$ $1$ .Simplifying the result from the product rule, we get $u = -x$ $2$ .Now we combine the derivatives of the two terms to find the derivative of the entire function $u = -x$ $3$ . The derivative of $u = -x$ $3$ with respect to $x$ is $u = -x$ $6$ .We can simplify the expression by combining the terms over the common denominator $\sqrt{a^2 - x^2}$ . The final derivative of $u = -x$ $3$ with respect to $x$ is $v = \sqrt{a^2 - x^2}$ $0$ .
Simplify Final Derivative: Now, let's find the derivative of the second term $-x \cdot \sqrt{a^2 - x^2}$ . We will use the product rule, which states that $\frac{d}{dx}(u\cdot v) = u' \cdot v + u \cdot v'$ , where $u = -x$ and $v = \sqrt{a^2 - x^2}$ .The derivative of $-x$ with respect to $x$ is $-1$ . Now we need to find the derivative of $\sqrt{a^2 - x^2}$ with respect to $x$ . We will use the chain rule again, where the outer function is the square root and the inner function is $(a^2 - x^2)$ .The derivative of $\frac{d}{dx}(u\cdot v) = u' \cdot v + u \cdot v'$ $0$ with respect to $\frac{d}{dx}(u\cdot v) = u' \cdot v + u \cdot v'$ $1$ is $\frac{d}{dx}(u\cdot v) = u' \cdot v + u \cdot v'$ $2$ . The derivative of $(a^2 - x^2)$ with respect to $x$ is $\frac{d}{dx}(u\cdot v) = u' \cdot v + u \cdot v'$ $5$ . Applying the chain rule, the derivative of $\sqrt{a^2 - x^2}$ is $\frac{d}{dx}(u\cdot v) = u' \cdot v + u \cdot v'$ $7$ .Simplifying the derivative of $\sqrt{a^2 - x^2}$ , we get $\frac{d}{dx}(u\cdot v) = u' \cdot v + u \cdot v'$ $9$ .Applying the product rule to the second term $-x \cdot \sqrt{a^2 - x^2}$ , we get $u = -x$ $1$ .Simplifying the result from the product rule, we get $u = -x$ $2$ .Now we combine the derivatives of the two terms to find the derivative of the entire function $u = -x$ $3$ . The derivative of $u = -x$ $3$ with respect to $x$ is $u = -x$ $6$ .We can simplify the expression by combining the terms over the common denominator $\sqrt{a^2 - x^2}$ . The final derivative of $u = -x$ $3$ with respect to $x$ is $v = \sqrt{a^2 - x^2}$ $0$ .After simplifying the numerator, we notice that the $v = \sqrt{a^2 - x^2}$ $1$ terms cancel out, and we are left with $v = \sqrt{a^2 - x^2}$ $2$ . This is the final derivative of the function $u = -x$ $3$ with respect to $x$ .

$y=a^{2}\arcsin\left(\frac{x}{a}\right)-x\sqrt{a^{2}-x^{2}}\quad(a=\text{constant})$

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$y=a^{2}\arcsin\left(\frac{x}{a}\right)-x\sqrt{a^{2}-x^{2}}\quad(a=\text{constant})$