Algebra
Calculus
Trigonometry
Matrix
Question
Function
Find the first partial derivative with respect to d
Find the first partial derivative with respect to x
∂d∂y=−51x2
Evaluate
y=−d×51x2−2x−6
Use the commutative property to reorder the terms
y=−51dx2−2x−6
Find the first partial derivative by treating the variable x as a constant and differentiating with respect to d
∂d∂y=∂d∂(−51dx2−2x−6)
Use differentiation rule ∂x∂(f(x)±g(x))=∂x∂(f(x))±∂x∂(g(x))
∂d∂y=−∂d∂(51dx2)−∂d∂(2x)−∂d∂(6)
Evaluate
More Steps
Evaluate
∂d∂(51dx2)
Use differentiation rule ∂x∂(cf(x))=c×∂x∂(f(x))
51x2×∂d∂(d)
Use ∂x∂xn=nxn−1 to find derivative
51x2×1
Multiply the terms
51x2
∂d∂y=−51x2−∂d∂(2x)−∂d∂(6)
Use ∂x∂(c)=0 to find derivative
∂d∂y=−51x2−0−∂d∂(6)
Use ∂x∂(c)=0 to find derivative
∂d∂y=−51x2−0−0
Solution
∂d∂y=−51x2
Show Solution
Solve the equation
Solve for x
Solve for d
Solve for y
x=−d5+25−30d−5dyx=d−5+25−30d−5dy
Evaluate
y=−d×51x2−2x−6
Use the commutative property to reorder the terms
y=−51dx2−2x−6
Swap the sides of the equation
−51dx2−2x−6=y
Move the expression to the left side
−51dx2−2x−6−y=0
Multiply both sides
5(−51dx2−2x−6−y)=5×0
Calculate
−dx2−10x−30−5y=0
Substitute a=−d,b=−10 and c=−30−5y into the quadratic formula x=2a−b±b2−4ac
x=2(−d)10±(−10)2−4(−d)(−30−5y)
Simplify the expression
x=−2d10±(−10)2−4(−d)(−30−5y)
Simplify the expression
More Steps
Evaluate
(−10)2−4(−d)(−30−5y)
Multiply the terms
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Multiply the terms
4(−d)(−30−5y)
Use the commutative property to reorder the terms
−4d(−30−5y)
Multiply the terms
−(−120−20y)d
Multiply the terms
−(−120d−20yd)
Use the commutative property to reorder the terms
120d+20yd
(−10)2−(120d+20yd)
Rewrite the expression
102−(120d+20yd)
If a negative sign or a subtraction symbol appears outside parentheses, remove the parentheses and change the sign of every term within the parentheses
102−120d−20yd
Evaluate the power
100−120d−20yd
x=−2d10±100−120d−20yd
Simplify the radical expression
More Steps
Evaluate
100−120d−20yd
Factor the expression
20(5−6d−yd)
The root of a product is equal to the product of the roots of each factor
20×5−6d−yd
Evaluate the root
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Evaluate
20
Write the expression as a product where the root of one of the factors can be evaluated
4×5
Write the number in exponential form with the base of 2
22×5
The root of a product is equal to the product of the roots of each factor
22×5
Reduce the index of the radical and exponent with 2
25
25×5−6d−yd
Calculate the product
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Evaluate
5×5−6d−yd
The product of roots with the same index is equal to the root of the product
5(5−6d−yd)
Calculate the product
25−30d−5yd
225−30d−5yd
Simplify
225−30d−5dy
x=−2d10±225−30d−5dy
Separate the equation into 2 possible cases
x=−2d10+225−30d−5dyx=−2d10−225−30d−5dy
Simplify the expression
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Evaluate
x=−2d10+225−30d−5dy
Divide the terms
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Evaluate
−2d10+225−30d−5dy
Rewrite the expression
−2d2(5+25−30d−5dy)
Cancel out the common factor 2
−d5+25−30d−5dy
Use b−a=−ba=−ba to rewrite the fraction
−d5+25−30d−5dy
x=−d5+25−30d−5dy
x=−d5+25−30d−5dyx=−2d10−225−30d−5dy
Solution
More Steps
Evaluate
x=−2d10−225−30d−5dy
Divide the terms
More Steps
Evaluate
−2d10−225−30d−5dy
Rewrite the expression
−2d2(5−25−30d−5dy)
Cancel out the common factor 2
−d5−25−30d−5dy
Use b−a=−ba=−ba to rewrite the fraction
−d5−25−30d−5dy
Rewrite the expression
d−5+25−30d−5dy
x=d−5+25−30d−5dy
x=−d5+25−30d−5dyx=d−5+25−30d−5dy
Show Solution